L11a424

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v 2 u 2 -5 v u 2 - v 2 w u 2 + 3 v w u 2 -3 w u 2 + 3 u 2 -5 v 2 u + 8 v u + 3 v 2 w u -8 v w u + 5 w u -3 u + 3 v 2 -3 v -3 v 2 w + 5 v w -2 w + 1$ (db)
Jones polynomial	$- q 2 + 5 q -10 + 15 q -1 -20 q -2 + 22 q -3 -20 q -4 + 18 q -5 -11 q -6 + 7 q -7 -2 q -8 + q -9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a 10 z -2 -2 a 8 z -2 -4 a 8 + 6 z 2 a 6 + a 6 z -2 + 5 a 6 -4 z 4 a 4 -5 z 2 a 4 -2 a 4 + z 6 a 2 + 2 z 4 a 2 + 4 z 2 a 2 + a 2 - z 4$ (db)
Kauffman polynomial	$z 6 a 10 -4 z 4 a 10 + 6 z 2 a 10 + a 10 z -2 -4 a 10 + 2 z 7 a 9 -4 z 5 a 9 + 4 z a 9 -2 a 9 z -1 + 3 z 8 a 8 -3 z 6 a 8 -5 z 4 a 8 + 9 z 2 a 8 + 2 a 8 z -2 -6 a 8 + 3 z 9 a 7 -7 z 5 a 7 + 5 z 3 a 7 -2 a 7 z -1 + z 10 a 6 + 11 z 8 a 6 -26 z 6 a 6 + 19 z 4 a 6 -2 z 2 a 6 + a 6 z -2 -3 a 6 + 8 z 9 a 5 - z 7 a 5 -24 z 5 a 5 + 24 z 3 a 5 -6 z a 5 + z 10 a 4 + 18 z 8 a 4 -37 z 6 a 4 + 21 z 4 a 4 -2 z 2 a 4 - a 4 + 5 z 9 a 3 + 11 z 7 a 3 -37 z 5 a 3 + 24 z 3 a 3 -2 z a 3 + 10 z 8 a 2 -10 z 6 a 2 -4 z 4 a 2 + 3 z 2 a 2 - a 2 + 10 z 7 a -15 z 5 a + 5 z 3 a + 5 z 6 -5 z 4 + z 5 a -1$ (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 =

-2 is the signature of L11a424. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2$	${\mathbb Z}^{3}$
$r = -5$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -4$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{8}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 0$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$