L11a459

(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_12,15,5,16 X_22,20,13,19 X_18,8,19,7 X_10,14,11,13 X_8,21,9,22 X_20,9,21,10 X_2,11,3,12 X_4,18,1,17
Gauss code	{1, -10, 2, -11}, {3, -1, 6, -8, 9, -7, 10, -4}, {7, -2, 4, -3, 11, -6, 5, -9, 8, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 3 u 3 -2 v 2 u 3 + v u 3 - v 3 w u 3 + 2 v 2 w u 3 - v w u 3 -2 v 3 u 2 + 5 v 2 u 2 -4 v u 2 + 2 v 3 w u 2 -5 v 2 w u 2 + 5 v w u 2 - w u 2 + u 2 + v 3 u -5 v 2 u + 5 v u - v 3 w u + 4 v 2 w u -5 v w u + 2 w u -2 u + v 2 -2 v - v 2 w + 2 v w - w + 1$ (db)
Jones polynomial	$- q 5 + 4 q 4 -8 q 3 + 13 q 2 -18 q + 22-20 q -1 + 19 q -2 -13 q -3 + 9 q -4 -4 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 8 -2 a 2 z 6 - z 6 a -2 + 5 z 6 + a 4 z 4 -7 a 2 z 4 -3 z 4 a -2 + 9 z 4 + 2 a 4 z 2 -7 a 2 z 2 -2 z 2 a -2 + 6 z 2 + a 4 -3 a 2 + 2 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$2 a 2 z 10 + 2 z 10 + 6 a 3 z 9 + 12 a z 9 + 6 z 9 a -1 + 7 a 4 z 8 + 11 a 2 z 8 + 8 z 8 a -2 + 12 z 8 + 4 a 5 z 7 -9 a 3 z 7 -24 a z 7 -4 z 7 a -1 + 7 z 7 a -3 + a 6 z 6 -17 a 4 z 6 -39 a 2 z 6 -12 z 6 a -2 + 4 z 6 a -4 -37 z 6 -9 a 5 z 5 -2 a 3 z 5 + 14 a z 5 -5 z 5 a -1 -11 z 5 a -3 + z 5 a -5 -2 a 6 z 4 + 12 a 4 z 4 + 40 a 2 z 4 + 7 z 4 a -2 -6 z 4 a -4 + 39 z 4 + 4 a 5 z 3 + 4 a 3 z 3 + a z 3 + 6 z 3 a -1 + 4 z 3 a -3 - z 3 a -5 + a 6 z 2 -7 a 4 z 2 -18 a 2 z 2 -3 z 2 a -2 + z 2 a -4 -14 z 2 -3 a 3 z -3 a z + 3 a 4 + 5 a 2 + 3 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 =

0 is the signature of L11a459. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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