L11a164

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,5,15,6 X_20,16,21,15 X_18,12,19,11 X_12,20,13,19 X_22,18,7,17 X_16,22,17,21 X₆₇₁₈ X_4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 6, -7, 11, -4, 5, -9, 8, -6, 7, -5, 9, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 6 + 2 v 2 u 5 -2 v u 5 -3 v 2 u 4 + 4 v u 4 -2 u 4 + 3 v 2 u 3 -5 v u 3 + 3 u 3 -2 v 2 u 2 + 4 v u 2 -3 u 2 -2 v u + 2 u -1$ (db)
Jones polynomial	$q^{9/2}-2 q^{7/2}+5 q^{5/2}-8 q^{3/2}+10 \sqrt{q}-\frac{13}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- a z 9 + a 3 z 7 -8 a z 7 + z 7 a -1 + 6 a 3 z 5 -25 a z 5 + 6 z 5 a -1 + 13 a 3 z 3 -37 a z 3 + 13 z 3 a -1 + 11 a 3 z -25 a z + 11 z a -1 + 3 a 3 z -1 -5 a z -1 + 2 a -1 z -1$ (db)
Kauffman polynomial	$- a 2 z 10 - z 10 -3 a 3 z 9 -6 a z 9 -3 z 9 a -1 -3 a 4 z 8 -3 a 2 z 8 -3 z 8 a -2 -3 z 8 -3 a 5 z 7 + 9 a 3 z 7 + 24 a z 7 + 10 z 7 a -1 -2 z 7 a -3 -2 a 6 z 6 + 5 a 4 z 6 + 17 a 2 z 6 + 9 z 6 a -2 - z 6 a -4 + 20 z 6 - a 7 z 5 + 6 a 5 z 5 -20 a 3 z 5 -49 a z 5 -16 z 5 a -1 + 6 z 5 a -3 + 4 a 6 z 4 -4 a 4 z 4 -34 a 2 z 4 -7 z 4 a -2 + 4 z 4 a -4 -37 z 4 + 3 a 7 z 3 -4 a 5 z 3 + 22 a 3 z 3 + 51 a z 3 + 19 z 3 a -1 -3 z 3 a -3 - a 6 z 2 + 2 a 4 z 2 + 23 a 2 z 2 + 3 z 2 a -2 -4 z 2 a -4 + 27 z 2 -2 a 7 z + 2 a 5 z -13 a 3 z -28 a z -11 z a -1 -5 a 2 + a -4 -5 + 3 a 3 z -1 + 5 a z -1 + 2 a -1 z -1$ (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 L11a164. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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