L11a317

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

Planar diagram presentation	X_10,1,11,2 X_20,11,21,12 X_6,9,7,10 X_16,7,17,8 X_8,15,1,16 X_22,17,9,18 X_12,4,13,3 X_18,6,19,5 X_4,14,5,13 X_14,21,15,22 X_2,20,3,19
Gauss code	{1, -11, 7, -9, 8, -3, 4, -5}, {3, -1, 2, -7, 9, -10, 5, -4, 6, -8, 11, -2, 10, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 5 + v u 5 - v 3 u 4 + 5 v 2 u 4 -5 v u 4 + u 4 + 3 v 3 u 3 -12 v 2 u 3 + 12 v u 3 -3 u 3 -3 v 3 u 2 + 12 v 2 u 2 -12 v u 2 + 3 u 2 + v 3 u -5 v 2 u + 5 v u - u + v 2 - v$ (db)
Jones polynomial	$-q^{7/2}+5 q^{5/2}-12 q^{3/2}+19 \sqrt{q}-\frac{26}{\sqrt{q}}+\frac{28}{q^{3/2}}-\frac{29}{q^{5/2}}+\frac{24}{q^{7/2}}-\frac{17}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 3 z 7 + a z 7 - a 5 z 5 + 3 a 3 z 5 + 2 a z 5 - z 5 a -1 -2 a 5 z 3 + 5 a 3 z 3 - z 3 a -1 -2 a 5 z + 5 a 3 z -3 a z - a 5 z -1 + 3 a 3 z -1 -2 a z -1$ (db)
Kauffman polynomial	$-4 a 4 z 10 -4 a 2 z 10 -9 a 5 z 9 -21 a 3 z 9 -12 a z 9 -8 a 6 z 8 -11 a 4 z 8 -19 a 2 z 8 -16 z 8 -4 a 7 z 7 + 16 a 5 z 7 + 40 a 3 z 7 + 8 a z 7 -12 z 7 a -1 - a 8 z 6 + 17 a 6 z 6 + 38 a 4 z 6 + 48 a 2 z 6 -5 z 6 a -2 + 23 z 6 + 8 a 7 z 5 -11 a 5 z 5 -26 a 3 z 5 + 9 a z 5 + 15 z 5 a -1 - z 5 a -3 + 2 a 8 z 4 -13 a 6 z 4 -32 a 4 z 4 -29 a 2 z 4 + 3 z 4 a -2 -9 z 4 -4 a 7 z 3 + 7 a 5 z 3 + 15 a 3 z 3 - a z 3 -5 z 3 a -1 - a 8 z 2 + 5 a 6 z 2 + 12 a 4 z 2 + 8 a 2 z 2 + 2 z 2 -4 a 5 z -9 a 3 z -5 a z - a 6 -3 a 4 -3 a 2 + a 5 z -1 + 3 a 3 z -1 + 2 a z -1$ (db)

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

-1 is the signature of L11a317. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -3$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -2$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = -1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15}$	${\mathbb Z}^{15}$
$r = 0$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{15}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$