L11a458

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_16,5,17,6 X_20,9,21,10 X_18,8,19,7 X_22,19,13,20 X_8,14,9,13 X_12,15,5,16 X_10,21,11,22 X_2,11,3,12 X_4,18,1,17
Gauss code	{1, -10, 2, -11}, {3, -1, 5, -7, 4, -9, 10, -8}, {7, -2, 8, -3, 11, -5, 6, -4, 9, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 3 u 3 -2 v 2 u 3 + v u 3 - v 3 w u 3 + 2 v 2 w u 3 - v w u 3 -2 v 3 u 2 + 7 v 2 u 2 -5 v u 2 + 2 v 3 w u 2 -7 v 2 w u 2 + 6 v w u 2 - w u 2 + u 2 + v 3 u -6 v 2 u + 7 v u - v 3 w u + 5 v 2 w u -7 v w u + 2 w u -2 u + v 2 -2 v - v 2 w + 2 v w - w + 1$ (db)
Jones polynomial	$q 3 -4 q 2 + 9 q -14 + 22 q -1 -24 q -2 + 26 q -3 -22 q -4 + 17 q -5 -11 q -6 + 5 q -7 - q -8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 2 z 8 + 2 a 4 z 6 -5 a 2 z 6 + z 6 - a 6 z 4 + 6 a 4 z 4 -10 a 2 z 4 + 3 z 4 - a 6 z 2 + 5 a 4 z 2 -8 a 2 z 2 + 3 z 2 + a 4 -3 a 2 + 2 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$2 a 4 z 10 + 2 a 2 z 10 + 8 a 5 z 9 + 14 a 3 z 9 + 6 a z 9 + 13 a 6 z 8 + 22 a 4 z 8 + 16 a 2 z 8 + 7 z 8 + 11 a 7 z 7 + 3 a 5 z 7 -17 a 3 z 7 -5 a z 7 + 4 z 7 a -1 + 5 a 8 z 6 -18 a 6 z 6 -53 a 4 z 6 -47 a 2 z 6 + z 6 a -2 -16 z 6 + a 9 z 5 -15 a 7 z 5 -24 a 5 z 5 -7 a 3 z 5 -8 a z 5 -9 z 5 a -1 -4 a 8 z 4 + 6 a 6 z 4 + 40 a 4 z 4 + 45 a 2 z 4 -2 z 4 a -2 + 13 z 4 + 4 a 7 z 3 + 13 a 5 z 3 + 12 a 3 z 3 + 8 a z 3 + 5 z 3 a -1 - a 6 z 2 -14 a 4 z 2 -20 a 2 z 2 + z 2 a -2 -6 z 2 -3 a 3 z -3 a z + 3 a 4 + 5 a 2 + 3 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 L11a458. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -4$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{8}$
$r = -3$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -2$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 0$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{14}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$