L11a423

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_20,10,21,9 X_16,8,17,7 X_18,14,19,13 X_14,20,15,19 X_22,16,11,15 X_10,18,5,17 X_8,22,9,21 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, 3, -8}, {11, -2, 5, -6, 7, -4, 8, -5, 6, -3, 9, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v u 4 - v w u 4 + w u 4 - u 4 + 3 v 2 u 3 -6 v u 3 - v 2 w u 3 + 4 v w u 3 -3 w u 3 + 2 u 3 -4 v 2 u 2 + 7 v u 2 + 2 v 2 w u 2 -7 v w u 2 + 4 w u 2 -2 u 2 + 3 v 2 u -4 v u -2 v 2 w u + 6 v w u -3 w u + u - v 2 + v + v 2 w -2 v w$ (db)
Jones polynomial	$- q 8 + 4 q 7 -10 q 6 + 16 q 5 -21 q 4 + 25 q 3 -23 q 2 + 21 q -14 + 9 q -1 -3 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + z 6 a -4 - z 4 a -2 + 2 z 4 a -4 - z 4 a -6 -2 z 4 + a 2 z 2 -8 z 2 a -2 + 6 z 2 a -4 - z 2 a -6 - z 2 + a 2 -11 a -2 + 9 a -4 -2 a -6 + 3-5 a -2 z -2 + 4 a -4 z -2 - a -6 z -2 + 2 z -2$ (db)
Kauffman polynomial	$2 z 10 a -2 + 2 z 10 a -4 + 5 z 9 a -1 + 13 z 9 a -3 + 8 z 9 a -5 + 14 z 8 a -2 + 20 z 8 a -4 + 12 z 8 a -6 + 6 z 8 + 3 a z 7 -14 z 7 a -3 -2 z 7 a -5 + 9 z 7 a -7 + a 2 z 6 -43 z 6 a -2 -54 z 6 a -4 -22 z 6 a -6 + 4 z 6 a -8 -14 z 6 -6 a z 5 -22 z 5 a -1 -21 z 5 a -3 -20 z 5 a -5 -14 z 5 a -7 + z 5 a -9 -3 a 2 z 4 + 48 z 4 a -2 + 52 z 4 a -4 + 19 z 4 a -6 -4 z 4 a -8 + 16 z 4 + 3 a z 3 + 29 z 3 a -1 + 45 z 3 a -3 + 29 z 3 a -5 + 9 z 3 a -7 - z 3 a -9 + 3 a 2 z 2 -38 z 2 a -2 -29 z 2 a -4 -9 z 2 a -6 -15 z 2 -18 z a -1 -33 z a -3 -19 z a -5 -4 z a -7 - a 2 + 20 a -2 + 15 a -4 + 3 a -6 + 8 + 5 a -1 z -1 + 9 a -3 z -1 + 5 a -5 z -1 + a -7 z -1 -5 a -2 z -2 -4 a -4 z -2 - a -6 z -2 -2 z -2$ (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 =

2 is the signature of L11a423. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$
$r = -4$	${\mathbb Z}$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 0$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 2$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{13}$
$r = 3$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 4$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$