L11a457

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 3 + 3 v u 3 + v 2 w u 3 -2 v w u 3 + w u 3 - u 3 - v 3 u 2 + 5 v 2 u 2 -6 v u 2 -4 v 2 w u 2 + 6 v w u 2 -2 w u 2 + u 2 + 2 v 3 u -6 v 2 u + 4 v u - v 3 w u + 6 v 2 w u -5 v w u + w u - v 3 + 2 v 2 - v + v 3 w -3 v 2 w + 2 v w$ (db)
Jones polynomial	$q 3 -4 q 2 + 10 q -15 + 21 q -1 -22 q -2 + 23 q -3 -19 q -4 + 14 q -5 -7 q -6 + 3 q -7 - q -8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 8 + 3 z 2 a 6 + 2 a 6 -3 z 4 a 4 -2 z 2 a 4 + a 4 z -2 + a 4 + z 6 a 2 -3 z 2 a 2 -2 a 2 z -2 -5 a 2 -2 z 4 + z -2 + 3 + z 2 a -2$ (db)
Kauffman polynomial	$2 a 4 z 10 + 2 a 2 z 10 + 6 a 5 z 9 + 13 a 3 z 9 + 7 a z 9 + 7 a 6 z 8 + 15 a 4 z 8 + 16 a 2 z 8 + 8 z 8 + 5 a 7 z 7 -3 a 5 z 7 -21 a 3 z 7 -9 a z 7 + 4 z 7 a -1 + 3 a 8 z 6 -9 a 6 z 6 -45 a 4 z 6 -53 a 2 z 6 + z 6 a -2 -19 z 6 + a 9 z 5 -5 a 7 z 5 -6 a 5 z 5 + 3 a 3 z 5 -5 a z 5 -8 z 5 a -1 -5 a 8 z 4 + 6 a 6 z 4 + 52 a 4 z 4 + 59 a 2 z 4 -2 z 4 a -2 + 16 z 4 -2 a 9 z 3 + 10 a 5 z 3 + 10 a 3 z 3 + 6 a z 3 + 4 z 3 a -1 + 3 a 8 z 2 -30 a 4 z 2 -39 a 2 z 2 + z 2 a -2 -11 z 2 + a 9 z + a 7 z -3 a 5 z -8 a 3 z -5 a z - a 8 + 9 a 4 + 13 a 2 + 6 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 L11a457. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{6}$
$r = -3$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 0$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{13}$
$r = 1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$