L11a322

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 5 + v u 5 - v 3 u 4 + 4 v 2 u 4 -4 v u 4 + u 4 + 3 v 3 u 3 -8 v 2 u 3 + 8 v u 3 -3 u 3 -3 v 3 u 2 + 8 v 2 u 2 -8 v u 2 + 3 u 2 + v 3 u -4 v 2 u + 4 v u - u + v 2 - v$ (db)
Jones polynomial	$-q^{11/2}+4 q^{9/2}-8 q^{7/2}+13 q^{5/2}-19 q^{3/2}+21 \sqrt{q}-\frac{22}{\sqrt{q}}+\frac{19}{q^{3/2}}-\frac{15}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 + z 7 a -1 - a 3 z 5 + 3 a z 5 + 3 z 5 a -1 - z 5 a -3 -2 a 3 z 3 + 3 a z 3 + 2 z 3 a -1 -2 z 3 a -3 - a 3 z + 2 a z - z a -1 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-3 z 10 a -2 -3 z 10 -9 a z 9 -15 z 9 a -1 -6 z 9 a -3 -14 a 2 z 8 -2 z 8 a -2 -4 z 8 a -4 -12 z 8 -14 a 3 z 7 + 6 a z 7 + 41 z 7 a -1 + 20 z 7 a -3 - z 7 a -5 -9 a 4 z 6 + 21 a 2 z 6 + 28 z 6 a -2 + 14 z 6 a -4 + 44 z 6 -4 a 5 z 5 + 20 a 3 z 5 + 17 a z 5 -30 z 5 a -1 -20 z 5 a -3 + 3 z 5 a -5 - a 6 z 4 + 7 a 4 z 4 -7 a 2 z 4 -31 z 4 a -2 -14 z 4 a -4 -32 z 4 + a 5 z 3 -11 a 3 z 3 -14 a z 3 + 6 z 3 a -1 + 6 z 3 a -3 -2 z 3 a -5 -2 a 4 z 2 + 8 z 2 a -2 + 3 z 2 a -4 + 7 z 2 + 2 a 3 z + 4 a z + 2 z a -1 + 1- a z -1 - a -1 z -1$ (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 L11a322. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{12}$
$r = 1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$