# Lexicographic degrees of two-bridge knots with 12 crossings

E. Brugallé, P. -V. Koseleff, D. Pecker

The two-brige knots are knots that admit trigonal form diagrams. They admit a polynomial parametrisation of degree $(3,b,c)$, where $3 < b < c$.
We proved in [BKP3]that the lexicographic degree of a two-bridge knot with $N \leq 11$ crossings or fewer, is $(3,b,c)$ where $b+c=3N$, see also details. When $N\geq 12$, we have $b+c \geq 3N$.
We list here the degree $b$ corresponding to the lexicographic degree $(3,b,c)$ for two-bridge knots with 12 crossings.
For each knot we show that there exists a diagram with $s$ sign changes in its Gauss sequence, such that there exists $c$ such that $b+c=3N=36$.
We list all (simple) diagrams that could have a smaller degree if the braid condition hold (quasipositity of the associated braid, see [BKP2]).

NameSchub.Conway Not.Lex. Deg.Details
$12a_{38}$ 71/33 $C(2,6,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 56 diagrams of degree $(3,16,20)$. For example $\quad C(2,6,1,2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(2,6,1,1,2)$ $b \geq 16$ $\quad D(2,7,2,2)$ $b \geq 17$ $\quad D(3,2,6,2)$ $b \geq 19$ $\quad D(2,5,1,2,1,2)$ $b \geq 16$
$12a_{169}$ 49/23 $C(2,7,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 56 diagrams of degree $(3,16,20)$. For example $\quad C(2,6,1,-2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(2,7,1,2)$ $b \geq 16$ $\quad D(3,8,2)$ $b \geq 19$ $\quad D(2,6,1,2,2)$ $b \geq 16$
$12a_{197}$ 69/32 $C(2,6,2,2)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 149 diagrams of degree $(3,17,19)$. For example $\quad C(2,7,-1,-2,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 4 simple diagrams with 15 crossings or fewer $\quad D(2,6,2,2)$ $b \geq 17$ $\quad D(2,5,1,3,2)$ $b \geq 17$ $\quad D(3,2,7,2)$ $b \geq 19$ $\quad D(2,4,1,2,3,2)$ $b \geq 17$
$12a_{204}$ 173/76 $C(2,3,1,1,1,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,14,22)$ 7 diagrams of degree $(3,14,22)$. For example $\quad C(2,3,1,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,3,1,1,1,1,1,2)$ $b \geq 14$
$12a_{206}$ 105/47 $C(2,4,3,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 57 diagrams of degree $(3,16,20)$. For example $\quad C(2,4,2,1,-2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(2,4,3,1,2)$ $b \geq 16$ $\quad D(3,4,4,2)$ $b \geq 19$ $\quad D(2,3,1,4,1,2)$ $b \geq 16$ $\quad D(2,4,2,1,2,2)$ $b \geq 16$
$12a_{221}$ 169/66 $C(2,1,1,3,1,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,14,22)$ 8 diagrams of degree $(3,14,22)$. For example $\quad C(2,1,1,3,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,1,1,3,1,1,1,2)$ $b \geq 14$
$12a_{226}$ 181/75 $C(2,2,2,2,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 52 diagrams of degree $(3,16,20)$. For example $\quad C(2,2,1,1,-3,-1,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 6 simple diagrams with 13 crossings or fewer $\quad D(2,2,3,2,2,2)$ $b \geq 19$ $\quad D(2,2,2,2,1,1,2)$ $b \geq 16$ $\quad D(3,2,2,2,2,2)$ $b \geq 19$ $\quad D(2,1,1,3,2,1,1,2)$ $b \geq 16$ $\quad D(2,2,1,1,3,1,1,2)$ $b \geq 16$ $\quad D(2,2,2,1,1,2,1,2)$ $b \geq 16$
$12a_{239}$ 87/40 $C(2,5,1,2,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(2,5,1,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,5,1,2,2)$ $b \geq 14$
$12a_{241}$ 127/57 $C(2,4,2,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 51 diagrams of degree $(3,16,20)$. For example $\quad C(2,4,2,1,2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(2,4,2,1,1,2)$ $b \geq 16$ $\quad D(2,4,3,2,2)$ $b \geq 16$ $\quad D(3,2,2,4,2)$ $b \geq 19$ $\quad D(2,3,1,3,1,1,2)$ $b \geq 16$ $\quad D(2,4,1,1,2,1,2)$ $b \geq 16$
$12a_{243}$ 133/60 $C(2,4,1,1,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(2,4,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,4,1,1,1,1,2)$ $b \geq 14$
$12a_{247}$ 163/71 $C(2,3,2,1,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 3 diagrams of degree $(3,14,22)$. For example $\quad C(2,2,1,-3,-1,-1,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,3,2,1,1,1,2)$ $b \geq 16$
$12a_{251}$ 159/59 $C(2,1,2,3,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 5 diagrams of degree $(3,14,22)$. For example $\quad C(2,1,2,3,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,1,2,3,1,1,2)$ $b \geq 14$
$12a_{254}$ 97/23 $C(4,4,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 59 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,-5,-1,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(4,4,1,1,2)$ $b \geq 16$ $\quad D(4,4,2,3)$ $b \geq 19$ $\quad D(4,5,2,2)$ $b \geq 16$ $\quad D(4,3,1,2,1,2)$ $b \geq 16$
$12a_{255}$ 107/28 $C(3,1,4,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(3,1,4,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,4,1,1,2)$ $b \geq 14$
$12a_{257}$ 191/80 $C(2,2,1,1,2,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 7 diagrams of degree $(3,14,22)$. For example $\quad C(2,2,1,1,2,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,2,1,1,2,1,1,2)$ $b \geq 14$
$12a_{259}$ 115/52 $C(2,4,1,2,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(2,4,1,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,4,1,2,1,2)$ $b \geq 14$
$12a_{300}$ 155/68 $C(2,3,1,1,2,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(2,3,1,1,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,3,1,1,2,1,2)$ $b \geq 14$
$12a_{302}$ 147/61 $C(2,2,2,3,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 50 diagrams of degree $(3,16,20)$. For example $\quad C(2,2,1,1,-4,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(2,2,2,3,1,2)$ $b \geq 16$ $\quad D(3,4,2,2,2)$ $b \geq 19$ $\quad D(2,2,1,1,4,1,2)$ $b \geq 16$ $\quad D(2,2,2,2,1,2,2)$ $b \geq 16$
$12a_{303}$ 153/64 $C(2,2,1,1,3,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(2,2,1,1,3,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,2,1,1,3,1,2)$ $b \geq 14$
$12a_{306}$ 147/64 $C(2,3,2,1,2,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(2,3,2,1,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,3,2,1,2,2)$ $b \geq 14$
$12a_{307}$ 157/69 $C(2,3,1,1,1,2,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 4 diagrams of degree $(3,14,22)$. For example $\quad C(2,3,-2,-1,-3,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,3,1,1,1,2,2)$ $b \geq 16$
$12a_{330}$ 95/43 $C(2,4,1,3,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 4 diagrams of degree $(3,14,22)$. For example $\quad C(2,3,2,-1,-3,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,4,1,3,2)$ $b \geq 16$
$12a_{378}$ 127/45 $C(2,1,4,1,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(2,1,4,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,1,4,1,1,1,2)$ $b \geq 14$
$12a_{379}$ 71/17 $C(4,5,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 59 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,-6,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(4,5,1,2)$ $b \geq 16$ $\quad D(4,6,3)$ $b \geq 19$ $\quad D(4,4,1,2,2)$ $b \geq 16$
$12a_{380}$ 77/20 $C(3,1,5,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(3,1,5,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,5,1,2)$ $b \geq 14$
$12a_{384}$ 151/62 $C(2,2,3,2,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 3 diagrams of degree $(3,14,22)$. For example $\quad C(2,2,2,1,-3,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,2,3,2,1,2)$ $b \geq 16$
$12a_{385}$ 161/66 $C(2,2,3,1,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 4 diagrams of degree $(3,14,22)$. For example $\quad C(3,-2,-1,-3,-2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,2,3,1,1,1,2)$ $b \geq 16$
$12a_{406}$ 179/74 $C(2,2,2,1,1,2,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 4 diagrams of degree $(3,14,22)$. For example $\quad C(2,2,1,2,-1,-1,-2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,2,2,1,1,2,2)$ $b \geq 16$
$12a_{425}$ 81/37 $C(2,5,3,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 67 diagrams of degree $(3,16,20)$. For example $\quad C(2,4,-1,-4,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 1 simple diagrams with 13 crossings or fewer $\quad D(2,5,3,2)$ $b \geq 16$
$12a_{437}$ 149/65 $C(2,3,2,2,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 3 diagrams of degree $(3,14,22)$. For example $\quad C(2,2,1,-3,-2,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,3,2,2,1,2)$ $b \geq 16$
$12a_{447}$ 121/43 $C(2,1,4,2,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 7 diagrams of degree $(3,14,22)$. For example $\quad C(2,1,4,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,1,4,2,1,2)$ $b \geq 14$
$12a_{454}$ 103/27 $C(3,1,4,2,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 4 diagrams of degree $(3,14,22)$. For example $\quad C(3,1,3,1,-3,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,4,2,2)$ $b \geq 16$
$12a_{471}$ 85/38 $C(2,4,4,2)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 71 diagrams of degree $(3,17,19)$. For example $\quad C(2,5,-1,-4,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 1 simple diagrams with 15 crossings or fewer $\quad D(2,4,4,2)$ $b \geq 17$
$12a_{477}$ 169/70 $C(2,2,2,2,2,2)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 74 diagrams of degree $(3,17,19)$. For example $\quad C(2,-3,-3,1,1,2,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 5 simple diagrams with 15 crossings or fewer $\quad D(2,2,2,2,2,2)$ $b \geq 17$ $\quad D(2,3,2,3,2,2)$ $b \geq 19$ $\quad D(3,2,3,2,2,2)$ $b \geq 19$ $\quad D(3,2,3,1,1,3,2)$ $b \geq 19$ $\quad D(3,2,3,2,1,1,3)$ $b \geq 19$
$12a_{482}$ 93/22 $C(4,4,2,2)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 154 diagrams of degree $(3,17,19)$. For example $\quad C(3,1,-5,-3,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 6 simple diagrams with 15 crossings or fewer $\quad D(4,4,2,2)$ $b \geq 17$ $\quad D(4,3,1,3,2)$ $b \geq 17$ $\quad D(4,4,1,1,3)$ $b \geq 17$ $\quad D(4,5,2,3)$ $b \geq 17$ $\quad D(4,2,1,2,3,2)$ $b \geq 17$ $\quad D(4,1,1,2,2,3,2)$ $b \geq 19$
$12a_{497}$ 209/81 $C(2,1,1,2,1,1,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,1,1,2,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{498}$ 207/76 $C(2,1,2,1,1,1,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,1,2,1,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{499}$ 233/89 $C(2,1,1,1,1,1,1,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,13,23)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,1,1,1,1,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{500}$ 167/60 $C(2,1,3,1,1,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,14,22)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,1,3,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{501}$ 199/55 $C(3,1,1,1,1,1,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,14,22)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,1,1,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{502}$ 91/37 $C(2,2,5,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 58 diagrams of degree $(3,16,20)$. For example $\quad C(3,-1,-1,-5,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(2,2,5,1,2)$ $b \geq 16$ $\quad D(3,6,2,2)$ $b \geq 19$ $\quad D(2,2,4,1,2,2)$ $b \geq 16$
$12a_{506}$ 185/68 $C(2,1,2,1,1,2,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,1,2,1,1,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{508}$ 129/56 $C(2,3,3,2,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 3 diagrams of degree $(3,14,22)$. For example $\quad C(2,3,2,1,-3,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(2,3,3,2,2)$ $b \geq 16$
$12a_{510}$ 193/81 $C(2,2,1,1,1,1,2,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,2,1,1,1,1,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{511}$ 125/51 $C(2,2,4,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 48 diagrams of degree $(3,16,20)$. For example $\quad C(2,2,3,1,-2,-2,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(2,2,4,1,1,2)$ $b \geq 16$ $\quad D(3,2,4,2,2)$ $b \geq 19$ $\quad D(2,2,3,1,2,1,2)$ $b \geq 16$
$12a_{512}$ 151/64 $C(2,2,1,3,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,2,1,3,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{514}$ 187/79 $C(2,2,1,2,1,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,2,1,2,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{517}$ 145/52 $C(2,1,3,1,2,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,1,3,1,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{518}$ 157/34 $C(4,1,1,1,1,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,14,22)$ 7 diagrams of degree $(3,14,22)$. For example $\quad C(4,1,1,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,1,1,1,1,1,1,2)$ $b \geq 14$
$12a_{519}$ 111/25 $C(4,2,3,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 61 diagrams of degree $(3,16,20)$. For example $\quad C(4,1,1,-4,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(4,2,3,1,2)$ $b \geq 16$ $\quad D(4,2,4,3)$ $b \geq 19$ $\quad D(4,1,1,4,1,2)$ $b \geq 16$ $\quad D(4,2,2,1,2,2)$ $b \geq 16$
$12a_{520}$ 133/36 $C(3,1,2,3,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,2,3,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{521}$ 113/48 $C(2,2,1,4,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,2,1,4,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{522}$ 173/73 $C(2,2,1,2,2,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,2,1,2,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{528}$ 183/67 $C(2,1,2,1,2,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,1,2,1,2,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{532}$ 125/33 $C(3,1,3,1,2,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,3,1,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{533}$ 137/31 $C(4,2,2,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 49 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,-3,-2,-1,-2,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(4,2,2,1,1,2)$ $b \geq 16$ $\quad D(4,2,2,2,3)$ $b \geq 19$ $\quad D(4,2,3,2,2)$ $b \geq 16$ $\quad D(4,1,1,3,1,1,2)$ $b \geq 16$ $\quad D(4,2,1,1,2,1,2)$ $b \geq 16$
$12a_{534}$ 163/44 $C(3,1,2,2,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,2,2,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{535}$ 175/47 $C(3,1,2,1,1,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,2,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{536}$ 137/29 $C(4,1,2,1,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(4,1,2,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{537}$ 179/50 $C(3,1,1,2,1,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(3,-2,-1,-2,-1,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,1,2,1,1,1,2)$ $b \geq 14$
$12a_{538}$ 83/13 $C(6,2,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 62 diagrams of degree $(3,16,20)$. For example $\quad C(6,1,1,-2,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(6,2,1,1,2)$ $b \geq 16$ $\quad D(6,2,2,3)$ $b \geq 19$ $\quad D(6,3,2,2)$ $b \geq 16$ $\quad D(6,1,1,2,1,2)$ $b \geq 16$
$12a_{539}$ 145/44 $C(3,3,2,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 3 diagrams of degree $(3,14,22)$. For example $\quad C(3,2,1,-3,-1,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,3,2,1,1,2)$ $b \geq 16$
$12a_{540}$ 165/49 $C(3,2,1,2,1,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,2,1,2,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{541}$ 153/41 $C(3,1,2,1,2,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,2,1,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{545}$ 143/63 $C(2,3,1,2,2,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(2,3,1,2,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{549}$ 111/26 $C(4,3,1,2,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 4 diagrams of degree $(3,14,22)$. For example $\quad C(4,3,1,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,3,1,2,2)$ $b \geq 14$
$12a_{550}$ 149/34 $C(4,2,1,1,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 4 diagrams of degree $(3,14,22)$. For example $\quad C(4,2,1,1,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,2,1,1,1,1,2)$ $b \geq 14$
$12a_{551}$ 103/18 $C(5,1,2,1,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 7 diagrams of degree $(3,14,22)$. For example $\quad C(5,1,2,1,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(5,1,2,1,1,2)$ $b \geq 14$
$12a_{552}$ 131/30 $C(4,2,1,2,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 5 diagrams of degree $(3,14,22)$. For example $\quad C(4,2,1,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,2,1,2,1,2)$ $b \geq 14$
$12a_{579}$ 177/49 $C(3,1,1,1,1,2,1,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,1,1,1,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{580}$ 69/11 $C(6,3,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 56 diagrams of degree $(3,16,20)$. For example $\quad C(7,-1,-2,-2,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(6,3,1,2)$ $b \geq 16$ $\quad D(6,4,3)$ $b \geq 19$ $\quad D(6,2,1,2,2)$ $b \geq 16$
$12a_{581}$ 119/36 $C(3,3,3,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 3 diagrams of degree $(3,14,22)$. For example $\quad C(3,2,1,-4,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,3,3,1,2)$ $b \geq 16$
$12a_{582}$ 131/39 $C(3,2,1,3,1,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,2,1,3,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{583}$ 161/45 $C(3,1,1,2,1,2,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,1,2,1,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{584}$ 143/31 $C(4,1,1,1,1,2,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(4,1,1,1,1,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{585}$ 181/50 $C(3,1,1,1,1,1,2,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 7 diagrams of degree $(3,14,22)$. For example $\quad C(3,1,1,1,1,1,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,1,1,1,1,2,2)$ $b \geq 14$
$12a_{595}$ 139/30 $C(4,1,1,1,2,1,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(4,1,1,1,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,1,1,1,2,1,2)$ $b \geq 14$
$12a_{596}$ 81/14 $C(5,1,3,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(5,1,3,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(5,1,3,1,2)$ $b \geq 14$
$12a_{597}$ 123/26 $C(4,1,2,1,2,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(4,1,2,1,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,1,2,1,2,2)$ $b \geq 14$
$12a_{600}$ 109/25 $C(4,2,1,3,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(4,2,1,3,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{601}$ 127/34 $C(3,1,2,1,3,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 7 diagrams of degree $(3,14,22)$. For example $\quad C(3,1,2,1,3,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,2,1,3,2)$ $b \geq 14$
$12a_{643}$ 99/23 $C(4,3,3,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 63 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,-4,-3,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(4,3,3,2)$ $b \geq 16$ $\quad D(4,2,1,4,2)$ $b \geq 16$ $\quad D(4,3,2,1,3)$ $b \geq 16$
$12a_{644}$ 113/30 $C(3,1,3,3,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 3 diagrams of degree $(3,14,22)$. For example $\quad C(3,1,3,2,1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,3,3,2)$ $b \geq 16$
$12a_{649}$ 127/27 $C(4,1,2,2,1,2)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(4,1,2,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{650}$ 165/46 $C(3,1,1,2,2,1,2)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 5 diagrams of degree $(3,14,22)$. For example $\quad C(3,1,1,2,2,1,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,1,2,2,1,2)$ $b \geq 14$
$12a_{651}$ 97/17 $C(5,1,2,2,2)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(5,1,2,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{652}$ 155/46 $C(3,2,1,2,2,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 5 diagrams of degree $(3,14,22)$. For example $\quad C(3,2,1,2,2,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,2,1,2,2,2)$ $b \geq 14$
$12a_{682}$ 107/29 $C(3,1,2,4,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 5 diagrams of degree $(3,14,22)$. For example $\quad C(3,1,2,4,2)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,2,4,2)$ $b \geq 14$
$12a_{684}$ 135/41 $C(3,3,2,2,2)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 3 diagrams of degree $(3,14,22)$. For example $\quad C(3,2,1,-3,-2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,3,2,2,2)$ $b \geq 16$
$12a_{690}$ 89/20 $C(4,2,4,2)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 155 diagrams of degree $(3,17,19)$. For example $\quad C(3,-1,-3,-3,1,3)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 6 simple diagrams with 15 crossings or fewer $\quad D(4,2,4,2)$ $b \geq 17$ $\quad D(4,1,1,5,2)$ $b \geq 17$ $\quad D(4,2,3,1,3)$ $b \geq 17$ $\quad D(5,2,5,2)$ $b \geq 19$ $\quad D(4,2,2,1,2,3)$ $b \geq 17$ $\quad D(4,2,1,1,2,2,3)$ $b \geq 19$
$12a_{691}$ 77/12 $C(6,2,2,2)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 161 diagrams of degree $(3,17,19)$. For example $\quad C(6,3,-1,-2,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 5 simple diagrams with 15 crossings or fewer $\quad D(6,2,2,2)$ $b \geq 17$ $\quad D(6,1,1,3,2)$ $b \geq 17$ $\quad D(6,2,1,1,3)$ $b \geq 17$ $\quad D(6,3,2,3)$ $b \geq 17$ $\quad D(7,2,3,2)$ $b \geq 19$
$12a_{713}$ 139/39 $C(3,1,1,3,2,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 58 diagrams of degree $(3,16,20)$. For example $\quad C(3,-1,-1,-3,-1,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(3,1,1,3,2,2)$ $b \geq 16$ $\quad D(3,2,4,2,2)$ $b \geq 19$ $\quad D(4,2,3,2,2)$ $b \geq 16$ $\quad D(3,1,1,2,1,3,2)$ $b \geq 16$
$12a_{714}$ 107/19 $C(5,1,1,1,2,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 57 diagrams of degree $(3,16,20)$. For example $\quad C(6,-2,-1,-3,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(5,1,1,1,2,2)$ $b \geq 16$ $\quad D(5,1,2,3,2)$ $b \geq 16$ $\quad D(5,2,2,2,2)$ $b \geq 19$ $\quad D(6,2,1,2,2)$ $b \geq 16$ $\quad D(5,1,1,1,1,1,3)$ $b \geq 16$
$12a_{715}$ 169/50 $C(3,2,1,1,1,2,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 54 diagrams of degree $(3,16,20)$. For example $\quad C(3,2,1,2,-3,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 6 simple diagrams with 13 crossings or fewer $\quad D(3,2,1,1,1,2,2)$ $b \geq 16$ $\quad D(3,2,1,2,3,2)$ $b \geq 16$ $\quad D(3,2,2,2,2,2)$ $b \geq 19$ $\quad D(3,3,2,1,2,2)$ $b \geq 16$ $\quad D(3,1,1,2,1,1,2,2)$ $b \geq 16$ $\quad D(3,2,1,1,1,1,1,3)$ $b \geq 16$
$12a_{716}$ 43/5 $C(8,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 62 diagrams of degree $(3,16,20)$. For example $\quad C(8,1,2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(8,1,1,2)$ $b \geq 16$ $\quad D(8,2,3)$ $b \geq 19$ $\quad D(9,2,2)$ $b \geq 19$
$12a_{717}$ 89/28 $C(3,5,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 54 diagrams of degree $(3,16,20)$. For example $\quad C(2,-2,-1,-6,1,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(3,5,1,1,2)$ $b \geq 16$ $\quad D(3,5,2,3)$ $b \geq 16$ $\quad D(3,6,2,2)$ $b \geq 19$ $\quad D(3,4,1,2,1,2)$ $b \geq 16$
$12a_{718}$ 141/41 $C(3,2,3,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 54 diagrams of degree $(3,16,20)$. For example $\quad C(3,3,-1,-2,-2,3)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(3,2,3,1,1,2)$ $b \geq 16$ $\quad D(3,2,4,2,2)$ $b \geq 19$ $\quad D(3,1,1,4,1,1,2)$ $b \geq 16$ $\quad D(3,2,2,1,2,1,2)$ $b \geq 16$
$12a_{720}$ 113/21 $C(5,2,1,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 51 diagrams of degree $(3,16,20)$. For example $\quad C(5,2,1,2,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(5,2,1,1,1,2)$ $b \geq 16$ $\quad D(5,2,1,2,3)$ $b \geq 16$ $\quad D(5,2,2,2,2)$ $b \geq 19$ $\quad D(5,3,2,1,2)$ $b \geq 16$ $\quad D(5,1,1,2,1,1,2)$ $b \geq 16$
$12a_{721}$ 171/50 $C(3,2,2,1,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 51 diagrams of degree $(3,16,20)$. For example $\quad C(3,-3,2,1,-3,-3)$, 15 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 6 simple diagrams with 13 crossings or fewer $\quad D(3,2,2,1,1,1,2)$ $b \geq 16$ $\quad D(3,2,2,1,2,3)$ $b \geq 16$ $\quad D(3,2,2,2,2,2)$ $b \geq 19$ $\quad D(3,2,3,2,1,2)$ $b \geq 17$ $\quad D(3,1,1,3,1,1,1,2)$ $b \geq 16$ $\quad D(3,2,1,1,2,1,1,2)$ $b \geq 16$
$12a_{722}$ 29/3 $C(9,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 79 diagrams of degree $(3,16,20)$. For example $\quad C(8,1,-2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(9,1,2)$ $b \geq 16$ $\quad D(10,3)$ $b \geq 19$
$12a_{723}$ 63/20 $C(3,6,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 52 diagrams of degree $(3,16,20)$. For example $\quad C(2,1,-7,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(3,6,1,2)$ $b \geq 16$ $\quad D(3,5,1,2,2)$ $b \geq 16$
$12a_{724}$ 107/31 $C(3,2,4,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 57 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,1,-5,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(3,5,2,3)$ $b \geq 16$ $\quad D(3,2,4,1,2)$ $b \geq 16$ $\quad D(3,1,1,5,1,2)$ $b \geq 16$ $\quad D(3,2,3,1,2,2)$ $b \geq 16$
$12a_{726}$ 103/19 $C(5,2,2,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 57 diagrams of degree $(3,16,20)$. For example $\quad C(5,1,1,-3,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(5,2,2,1,2)$ $b \geq 16$ $\quad D(5,2,3,3)$ $b \geq 17$ $\quad D(5,1,1,3,1,2)$ $b \geq 16$ $\quad D(5,2,1,1,2,2)$ $b \geq 16$
$12a_{727}$ 157/46 $C(3,2,2,2,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 43 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,1,-3,-2,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(3,3,2,2,3)$ $b \geq 16$ $\quad D(3,2,2,2,1,2)$ $b \geq 16$ $\quad D(3,1,1,3,2,1,2)$ $b \geq 16$ $\quad D(3,2,1,1,3,1,2)$ $b \geq 16$ $\quad D(3,2,2,1,1,2,2)$ $b \geq 17$
$12a_{728}$ 133/29 $C(4,1,1,2,2,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 54 diagrams of degree $(3,16,20)$. For example $\quad C(4,1,1,2,1,1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(4,1,1,2,2,2)$ $b \geq 16$ $\quad D(4,2,3,2,2)$ $b \geq 16$ $\quad D(5,2,2,2,2)$ $b \geq 19$ $\quad D(4,1,1,1,1,3,2)$ $b \geq 16$ $\quad D(4,1,1,2,1,1,3)$ $b \geq 16$
$12a_{729}$ 167/46 $C(3,1,1,1,2,2,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 54 diagrams of degree $(3,16,20)$. For example $\quad C(4,-2,-1,-2,-2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(3,1,1,1,2,2,2)$ $b \geq 16$ $\quad D(3,1,2,3,2,2)$ $b \geq 16$ $\quad D(3,2,2,2,2,2)$ $b \geq 19$ $\quad D(4,2,1,2,2,2)$ $b \geq 16$ $\quad D(3,1,1,1,1,1,3,2)$ $b \geq 16$
$12a_{731}$ 105/22 $C(4,1,3,2,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 59 diagrams of degree $(3,16,20)$. For example $\quad C(4,1,2,1,-3,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(4,1,3,2,2)$ $b \geq 16$ $\quad D(5,4,2,2)$ $b \geq 19$ $\quad D(4,1,2,1,3,2)$ $b \geq 16$ $\quad D(4,1,3,1,1,3)$ $b \geq 16$
$12a_{732}$ 95/18 $C(5,3,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 61 diagrams of degree $(3,16,20)$. For example $\quad C(5,3,1,2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(5,3,1,1,2)$ $b \geq 16$ $\quad D(5,3,2,3)$ $b \geq 16$ $\quad D(5,4,2,2)$ $b \geq 19$ $\quad D(5,2,1,2,1,2)$ $b \geq 16$
$12a_{733}$ 73/14 $C(5,4,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 47 diagrams of degree $(3,16,20)$. For example $\quad C(5,3,1,-2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(5,4,1,2)$ $b \geq 16$ $\quad D(5,5,3)$ $b \geq 16$ $\quad D(5,3,1,2,2)$ $b \geq 16$
$12a_{736}$ 141/43 $C(3,3,1,1,2,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 54 diagrams of degree $(3,16,20)$. For example $\quad C(2,2,1,1,4,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(3,3,1,1,2,2)$ $b \geq 16$ $\quad D(3,3,2,3,2)$ $b \geq 16$ $\quad D(3,4,2,2,2)$ $b \geq 19$ $\quad D(3,2,1,2,1,2,2)$ $b \geq 16$ $\quad D(3,3,1,1,1,1,3)$ $b \geq 16$
$12a_{738}$ 119/37 $C(3,4,1,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 55 diagrams of degree $(3,16,20)$. For example $\quad C(2,1,-5,-1,-1,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(3,4,1,1,1,2)$ $b \geq 16$ $\quad D(3,4,1,2,3)$ $b \geq 16$ $\quad D(3,4,2,2,2)$ $b \geq 19$ $\quad D(3,5,2,1,2)$ $b \geq 16$ $\quad D(3,3,1,2,1,1,2)$ $b \geq 16$
$12a_{740}$ 113/35 $C(3,4,2,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 47 diagrams of degree $(3,16,20)$. For example $\quad C(2,1,-5,-2,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(3,4,2,1,2)$ $b \geq 16$ $\quad D(3,4,3,3)$ $b \geq 16$ $\quad D(3,3,1,3,1,2)$ $b \geq 16$ $\quad D(3,4,1,1,2,2)$ $b \geq 16$
$12a_{743}$ 79/12 $C(6,1,1,2,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 57 diagrams of degree $(3,16,20)$. For example $\quad C(5,1,-2,-1,-2,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(6,1,1,2,2)$ $b \geq 16$ $\quad D(6,2,3,2)$ $b \geq 17$ $\quad D(7,2,2,2)$ $b \geq 19$ $\quad D(6,1,1,1,1,3)$ $b \geq 16$
$12a_{744}$ 61/8 $C(7,1,1,1,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 56 diagrams of degree $(3,16,20)$. For example $\quad C(7,1,2,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(7,1,1,1,2)$ $b \geq 16$ $\quad D(7,1,2,3)$ $b \geq 16$ $\quad D(7,2,2,2)$ $b \geq 19$ $\quad D(8,2,1,2)$ $b \geq 17$
$12a_{745}$ 59/8 $C(7,2,1,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 33 diagrams of degree $(3,16,20)$. For example $\quad C(6,1,-3,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(7,2,1,2)$ $b \geq 16$ $\quad D(7,3,3)$ $b \geq 17$ $\quad D(7,1,1,2,2)$ $b \geq 17$
$12a_{758}$ 113/31 $C(3,1,1,1,4,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 59 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,2,-5,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(3,1,1,1,4,2)$ $b \geq 16$ $\quad D(3,1,2,5,2)$ $b \geq 16$ $\quad D(3,2,2,4,2)$ $b \geq 19$ $\quad D(4,2,1,4,2)$ $b \geq 16$
$12a_{759}$ 61/9 $C(6,1,3,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 61 diagrams of degree $(3,16,20)$. For example $\quad C(6,1,2,1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(6,1,3,2)$ $b \geq 16$ $\quad D(7,4,2)$ $b \geq 19$ $\quad D(6,1,2,1,3)$ $b \geq 16$
$12a_{760}$ 111/34 $C(3,3,1,3,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 58 diagrams of degree $(3,16,20)$. For example $\quad C(3,2,1,-2,-3,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(3,3,1,3,2)$ $b \geq 16$ $\quad D(3,4,4,2)$ $b \geq 19$ $\quad D(3,2,1,2,3,2)$ $b \geq 16$ $\quad D(3,3,1,2,1,3)$ $b \geq 16$
$12a_{761}$ 139/41 $C(3,2,1,1,3,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 54 diagrams of degree $(3,16,20)$. For example $\quad C(2,3,1,1,3,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 5 simple diagrams with 13 crossings or fewer $\quad D(3,2,1,1,3,2)$ $b \geq 16$ $\quad D(3,2,2,4,2)$ $b \geq 19$ $\quad D(3,3,2,3,2)$ $b \geq 16$ $\quad D(3,1,1,2,1,3,2)$ $b \geq 16$ $\quad D(3,2,1,1,2,1,3)$ $b \geq 16$
$12a_{762}$ 51/7 $C(7,3,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 66 diagrams of degree $(3,16,20)$. For example $\quad C(7,2,1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(7,3,2)$ $b \geq 16$ $\quad D(7,2,1,3)$ $b \geq 16$
$12a_{763}$ 97/30 $C(3,4,3,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 40 diagrams of degree $(3,16,20)$. For example $\quad C(3,-1,-2,-5,1,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(3,4,3,2)$ $b \geq 16$ $\quad D(3,3,1,4,2)$ $b \geq 17$ $\quad D(3,4,2,1,3)$ $b \geq 16$
$12a_{764}$ 133/39 $C(3,2,2,3,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 44 diagrams of degree $(3,16,20)$. For example $\quad C(2,2,1,-3,-3,1,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(3,2,2,3,2)$ $b \geq 16$ $\quad D(3,1,1,3,3,2)$ $b \geq 16$ $\quad D(3,2,1,1,4,2)$ $b \geq 16$ $\quad D(3,2,2,2,1,3)$ $b \geq 16$
$12a_{773}$ 91/20 $C(4,1,1,4,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 57 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,-2,-1,-4,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(4,1,1,4,2)$ $b \geq 16$ $\quad D(4,2,5,2)$ $b \geq 17$ $\quad D(5,2,4,2)$ $b \geq 19$ $\quad D(4,1,1,3,1,3)$ $b \geq 16$
$12a_{774}$ 89/16 $C(5,1,1,3,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 57 diagrams of degree $(3,16,20)$. For example $\quad C(4,1,-2,-1,-3,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(5,1,1,3,2)$ $b \geq 16$ $\quad D(5,2,4,2)$ $b \geq 19$ $\quad D(6,2,3,2)$ $b \geq 17$ $\quad D(5,1,1,2,1,3)$ $b \geq 16$
$12a_{775}$ 87/16 $C(5,2,3,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 33 diagrams of degree $(3,16,20)$. For example $\quad C(5,2,2,1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(5,2,3,2)$ $b \geq 16$ $\quad D(5,1,1,4,2)$ $b \geq 17$ $\quad D(5,2,2,1,3)$ $b \geq 16$
$12a_{791}$ 63/13 $C(4,1,5,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 63 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,-2,-5,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(4,1,5,2)$ $b \geq 16$ $\quad D(5,6,2)$ $b \geq 19$ $\quad D(4,1,4,1,3)$ $b \geq 16$
$12a_{792}$ 85/24 $C(3,1,1,5,2)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 57 diagrams of degree $(3,16,20)$. For example $\quad C(2,5,1,2,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(3,1,1,5,2)$ $b \geq 16$ $\quad D(3,2,6,2)$ $b \geq 19$ $\quad D(4,2,5,2)$ $b \geq 17$
$12a_{796}$ 57/11 $C(5,5,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 46 diagrams of degree $(3,16,20)$. For example $\quad C(6,-1,-3,-1,3)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(5,5,2)$ $b \geq 16$ $\quad D(5,4,1,3)$ $b \geq 16$
$12a_{797}$ 83/24 $C(3,2,5,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 25 diagrams of degree $(3,16,20)$. For example $\quad C(3,3,-1,-3,-1,3)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(3,2,5,2)$ $b \geq 16$ $\quad D(3,1,1,6,2)$ $b \geq 17$ $\quad D(3,2,4,1,3)$ $b \geq 17$
$12a_{802}$ 47/15 $C(3,7,2)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 30 diagrams of degree $(3,16,20)$. For example $\quad C(4,-2,1,4,1,-3)$, 15 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(3,7,2)$ $b \geq 16$ $\quad D(3,6,1,3)$ $b \geq 16$
$12a_{803}$ 21/2 $C(10,2)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 142 diagrams of degree $(3,17,19)$. For example $\quad C(3,-1,-7,-1,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 1 simple diagrams with 15 crossings or fewer $\quad D(10,2)$ $b \geq 17$
$12a_{1023}$ 127/29 $C(4,2,1,1,1,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 67 diagrams of degree $(3,16,20)$. For example $\quad C(4,-2,-1,-2,-4)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(4,2,1,1,1,3)$ $b \geq 16$ $\quad D(4,2,2,2,3)$ $b \geq 19$ $\quad D(4,3,2,1,3)$ $b \geq 16$ $\quad D(4,1,1,2,1,1,3)$ $b \geq 16$
$12a_{1024}$ 149/40 $C(3,1,2,1,1,1,3)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,2,1,1,1,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{1029}$ 81/19 $C(4,3,1,4)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 75 diagrams of degree $(3,16,20)$. For example $\quad C(4,2,-1,-2,-4)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(4,3,1,4)$ $b \geq 16$ $\quad D(5,4,4)$ $b \geq 19$
$12a_{1030}$ 91/19 $C(4,1,3,1,3)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(4,1,3,1,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{1033}$ 107/25 $C(4,3,1,1,3)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 64 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,-4,-1,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(4,3,1,1,3)$ $b \geq 16$ $\quad D(4,3,2,4)$ $b \geq 17$ $\quad D(4,4,2,3)$ $b \geq 19$ $\quad D(4,2,1,2,1,3)$ $b \geq 16$
$12a_{1034}$ 121/32 $C(3,1,3,1,1,3)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 8 diagrams of degree $(3,14,22)$. For example $\quad C(3,1,3,1,1,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,1,3,1,1,3)$ $b \geq 14$
$12a_{1039}$ 137/37 $C(3,1,2,2,1,3)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,1,2,2,1,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{1040}$ 115/26 $C(4,2,2,1,3)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 4 diagrams of degree $(3,14,22)$. For example $\quad C(4,2,2,1,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,2,2,1,3)$ $b \geq 14$
$12a_{1125}$ 101/23 $C(4,2,1,1,4)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 62 diagrams of degree $(3,16,20)$. For example $\quad C(4,1,1,3,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(4,2,1,1,4)$ $b \geq 16$ $\quad D(4,3,2,4)$ $b \geq 17$ $\quad D(5,2,2,4)$ $b \geq 19$
$12a_{1126}$ 119/26 $C(4,1,1,2,1,3)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 7 diagrams of degree $(3,14,22)$. For example $\quad C(4,1,1,2,1,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,1,1,2,1,3)$ $b \geq 14$
$12a_{1127}$ 97/22 $C(4,2,2,4)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 83 diagrams of degree $(3,17,19)$. For example $\quad C(3,1,-3,-3,1,3)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 2 simple diagrams with 15 crossings or fewer $\quad D(4,2,2,4)$ $b \geq 17$ $\quad D(5,2,3,4)$ $b \geq 19$
$12a_{1128}$ 59/9 $C(6,1,1,4)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 75 diagrams of degree $(3,16,20)$. For example $\quad C(5,1,-2,-1,-4)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(6,1,1,4)$ $b \geq 16$ $\quad D(6,2,5)$ $b \geq 19$ $\quad D(7,2,4)$ $b \geq 19$
$12a_{1129}$ 105/23 $C(4,1,1,3,3)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 60 diagrams of degree $(3,16,20)$. For example $\quad C(4,1,1,4,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(4,1,1,3,3)$ $b \geq 16$ $\quad D(4,2,4,3)$ $b \geq 19$ $\quad D(5,2,3,3)$ $b \geq 17$
$12a_{1130}$ 125/27 $C(4,1,1,1,2,3)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 63 diagrams of degree $(3,16,20)$. For example $\quad C(5,-2,-1,-2,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(4,1,1,1,2,3)$ $b \geq 16$ $\quad D(4,1,2,3,3)$ $b \geq 16$ $\quad D(4,2,2,2,3)$ $b \geq 19$ $\quad D(5,2,1,2,3)$ $b \geq 16$
$12a_{1131}$ 73/11 $C(6,1,1,1,3)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 64 diagrams of degree $(3,16,20)$. For example $\quad C(7,-2,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(6,1,1,1,3)$ $b \geq 16$ $\quad D(6,1,2,4)$ $b \geq 16$ $\quad D(6,2,2,3)$ $b \geq 19$ $\quad D(7,2,1,3)$ $b \geq 16$
$12a_{1132}$ 131/40 $C(3,3,1,1,1,3)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 5 diagrams of degree $(3,14,22)$. For example $\quad C(4,-2,-1,-3,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,3,1,1,1,3)$ $b \geq 16$
$12a_{1133}$ 159/47 $C(3,2,1,1,1,1,3)$ 13
 Chebyshev parametrisation of degree $(3,16,20)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,2,1,1,1,1,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{1134}$ 53/7 $C(7,1,1,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 76 diagrams of degree $(3,16,20)$. For example $\quad C(6,1,-2,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(7,1,1,3)$ $b \geq 16$ $\quad D(7,2,4)$ $b \geq 19$ $\quad D(8,2,3)$ $b \geq 19$
$12a_{1135}$ 103/32 $C(3,4,1,1,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 56 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,1,5,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(3,4,1,1,3)$ $b \geq 16$ $\quad D(3,5,2,3)$ $b \geq 16$ $\quad D(4,2,4,3)$ $b \geq 19$ $\quad D(3,3,1,2,1,3)$ $b \geq 16$
$12a_{1136}$ 147/43 $C(3,2,2,1,1,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 62 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,1,-3,-1,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(3,2,2,1,1,3)$ $b \geq 16$ $\quad D(4,2,2,2,3)$ $b \geq 19$ $\quad D(3,2,1,1,2,1,3)$ $b \geq 16$
$12a_{1138}$ 79/14 $C(5,1,1,1,4)$ 16
 Chebyshev parametrisation of degree $(3,16,20)$ 65 diagrams of degree $(3,16,20)$. For example $\quad C(6,-2,-1,-4)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(5,2,1,5)$ $b \geq 16$ $\quad D(5,1,1,1,4)$ $b \geq 16$ $\quad D(5,2,2,4)$ $b \geq 19$ $\quad D(6,2,1,4)$ $b \geq 16$
$12a_{1139}$ 101/18 $C(5,1,1,1,1,3)$ 14
 Chebyshev parametrisation of degree $(3,16,20)$ 7 diagrams of degree $(3,14,22)$. For example $\quad C(5,1,1,1,1,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(5,1,1,1,1,3)$ $b \geq 14$
$12a_{1140}$ 97/18 $C(5,2,1,1,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 57 diagrams of degree $(3,16,20)$. For example $\quad C(5,1,1,-2,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(5,2,1,1,3)$ $b \geq 16$ $\quad D(5,2,2,4)$ $b \geq 19$ $\quad D(5,3,2,3)$ $b \geq 16$ $\quad D(5,1,1,2,1,3)$ $b \geq 16$
$12a_{1145}$ 79/15 $C(5,3,1,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 64 diagrams of degree $(3,16,20)$. For example $\quad C(5,2,1,-2,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(5,3,1,3)$ $b \geq 16$ $\quad D(5,4,4)$ $b \geq 19$ $\quad D(5,2,1,2,3)$ $b \geq 16$
$12a_{1146}$ 117/34 $C(3,2,3,1,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 51 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,4,-1,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(3,2,3,1,3)$ $b \geq 16$ $\quad D(4,4,2,3)$ $b \geq 19$ $\quad D(3,2,2,1,2,3)$ $b \geq 16$
$12a_{1148}$ 73/23 $C(3,5,1,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 59 diagrams of degree $(3,16,20)$. For example $\quad C(3,1,6,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(3,5,1,3)$ $b \geq 16$ $\quad D(4,6,3)$ $b \geq 19$ $\quad D(3,4,1,2,3)$ $b \geq 16$
$12a_{1149}$ 35/4 $C(8,1,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 52 diagrams of degree $(3,16,20)$. For example $\quad C(7,1,-2,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(8,1,3)$ $b \geq 16$ $\quad D(9,4)$ $b \geq 19$
$12a_{1157}$ 39/5 $C(7,1,4)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 68 diagrams of degree $(3,16,20)$. For example $\quad C(7,2,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(7,1,4)$ $b \geq 16$ $\quad D(8,5)$ $b \geq 19$
$12a_{1158}$ 77/16 $C(4,1,4,3)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 5 diagrams of degree $(3,14,22)$. For example $\quad C(4,-1,-3,-1,-4)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,1,4,3)$ $b \geq 16$
$12a_{1159}$ 113/24 $C(4,1,2,2,3)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(4,1,2,2,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{1161}$ 75/14 $C(5,2,1,4)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(5,2,1,4)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(5,2,1,4)$ $b \geq 14$
$12a_{1162}$ 69/13 $C(5,3,4)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 54 diagrams of degree $(3,16,20)$. For example $\quad C(5,-1,-3,1,4)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(5,3,4)$ $b \geq 16$ $\quad D(5,2,1,5)$ $b \geq 16$
$12a_{1163}$ 103/24 $C(4,3,2,3)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 3 diagrams of degree $(3,14,22)$. For example $\quad C(4,2,1,-3,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(4,3,2,3)$ $b \geq 16$
$12a_{1165}$ 67/16 $C(4,5,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 36 diagrams of degree $(3,16,20)$. For example $\quad C(5,-1,-3,-1,4)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(4,5,3)$ $b \geq 16$ $\quad D(4,4,1,4)$ $b \geq 16$
$12a_{1166}$ 33/4 $C(8,4)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 172 diagrams of degree $(3,17,19)$. For example $\quad C(3,1,-7,-1,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 1 simple diagrams with 15 crossings or fewer $\quad D(8,4)$ $b \geq 17$
$12a_{1273}$ 61/11 $C(5,1,1,5)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 36 diagrams of degree $(3,16,20)$. For example $\quad C(5,1,2,-1,-4)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(5,1,1,5)$ $b \geq 16$ $\quad D(6,2,5)$ $b \geq 19$
$12a_{1274}$ 95/17 $C(5,1,1,2,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 58 diagrams of degree $(3,16,20)$. For example $\quad C(5,1,1,3,-1,-2)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 4 simple diagrams with 13 crossings or fewer $\quad D(5,1,1,2,3)$ $b \geq 16$ $\quad D(5,2,3,3)$ $b \geq 17$ $\quad D(6,2,2,3)$ $b \geq 19$ $\quad D(5,1,1,1,1,4)$ $b \geq 16$
$12a_{1275}$ 149/44 $C(3,2,1,1,2,3)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 2 diagrams of degree $(3,14,22)$. For example $\quad C(3,2,1,2,-1,-1,-3)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(3,2,1,1,2,3)$ $b \geq 16$
$12a_{1276}$ 75/13 $C(5,1,3,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 66 diagrams of degree $(3,16,20)$. For example $\quad C(5,1,2,1,-4)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 3 simple diagrams with 13 crossings or fewer $\quad D(5,1,3,3)$ $b \geq 16$ $\quad D(6,4,3)$ $b \geq 19$ $\quad D(5,1,2,1,4)$ $b \geq 16$
$12a_{1277}$ 121/37 $C(3,3,1,2,3)$ 13
 Chebyshev parametrisation of degree $(3,17,19)$ 1 diagrams of degree $(3,13,23)$. For example $\quad C(3,3,1,2,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 23$
$12a_{1278}$ 41/6 $C(6,1,5)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 65 diagrams of degree $(3,16,20)$. For example $\quad C(6,2,-1,-4)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(6,1,5)$ $b \geq 16$ $\quad D(7,6)$ $b \geq 19$
$12a_{1279}$ 67/10 $C(6,1,2,3)$ 14
 Chebyshev parametrisation of degree $(3,17,19)$ 6 diagrams of degree $(3,14,22)$. For example $\quad C(6,1,2,3)$, 12 crossings, 23 sign changes in the Gauss sequence, ie $c \leq 22$ Braid condition for 1 simple diagrams with 12 crossings or fewer $\quad D(6,1,2,3)$ $b \geq 14$
$12a_{1281}$ 109/33 $C(3,3,3,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 28 diagrams of degree $(3,16,20)$. For example $\quad C(4,-1,-3,1,2,3)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 1 simple diagrams with 13 crossings or fewer $\quad D(3,3,3,3)$ $b \geq 16$
$12a_{1282}$ 63/10 $C(6,3,3)$ 16
 Chebyshev parametrisation of degree $(3,17,19)$ 48 diagrams of degree $(3,16,20)$. For example $\quad C(6,2,1,-4)$, 13 crossings, 22 sign changes in the Gauss sequence, ie $c \leq 20$ Braid condition for 2 simple diagrams with 13 crossings or fewer $\quad D(6,3,3)$ $b \geq 16$ $\quad D(6,2,1,4)$ $b \geq 16$
$12a_{1287}$ 37/6 $C(6,6)$ 17
 Chebyshev parametrisation of degree $(3,17,19)$ 81 diagrams of degree $(3,17,19)$. For example $\quad C(7,-1,-3,-1,2)$, 14 crossings, 21 sign changes in the Gauss sequence, ie $c \leq 19$ Braid condition for 1 simple diagrams with 15 crossings or fewer $\quad D(6,6)$ $b \geq 17$