L11a353

(Knotscape image)

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

Planar diagram presentation	X_12,1,13,2 X₈₄₉₃ X_16,6,17,5 X_22,8,11,7 X_20,15,21,16 X_14,21,15,22 X_6,14,7,13 X_4,20,5,19 X_18,9,19,10 X_2,11,3,12 X_10,17,1,18
Gauss code	{1, -10, 2, -8, 3, -7, 4, -2, 9, -11}, {10, -1, 7, -6, 5, -3, 11, -9, 8, -5, 6, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 4 v 2 u 3 -5 v u 3 + 2 u 3 + 2 v 3 u 2 -10 v 2 u 2 + 11 v u 2 -3 u 2 -3 v 3 u + 11 v 2 u -10 v u + 2 u + 2 v 3 -5 v 2 + 4 v -1$ (db)
Jones polynomial	$q^{11/2}-4 q^{9/2}+10 q^{7/2}-17 q^{5/2}+21 q^{3/2}-25 \sqrt{q}+\frac{24}{\sqrt{q}}-\frac{21}{q^{3/2}}+\frac{15}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 3 a z 5 -4 z 5 a -1 + z 5 a -3 -3 a 3 z 3 + 7 a z 3 -9 z 3 a -1 + 2 z 3 a -3 + a 5 z -3 a 3 z + 6 a z -7 z a -1 + 3 z a -3 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-3 a 2 z 10 -3 z 10 -6 a 3 z 9 -17 a z 9 -11 z 9 a -1 -4 a 4 z 8 -7 a 2 z 8 -18 z 8 a -2 -21 z 8 - a 5 z 7 + 16 a 3 z 7 + 38 a z 7 + 4 z 7 a -1 -17 z 7 a -3 + 13 a 4 z 6 + 42 a 2 z 6 + 29 z 6 a -2 -10 z 6 a -4 + 68 z 6 + 3 a 5 z 5 -9 a 3 z 5 -9 a z 5 + 32 z 5 a -1 + 25 z 5 a -3 -4 z 5 a -5 -14 a 4 z 4 -43 a 2 z 4 -12 z 4 a -2 + 7 z 4 a -4 - z 4 a -6 -49 z 4 -3 a 5 z 3 -4 a 3 z 3 -14 a z 3 -29 z 3 a -1 -16 z 3 a -3 + 5 a 4 z 2 + 11 a 2 z 2 + z 2 a -2 -3 z 2 a -4 + 10 z 2 + a 5 z + 2 a 3 z + 6 a z + 10 z a -1 + 5 z a -3 + 1- a z -1 - a -1 z -1$ (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 L11a353. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 0$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{14}$
$r = 1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$