L11a456

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 - v u 3 - v 2 w u 3 + 2 v w u 3 - w u 3 + v 3 u 2 -4 v 2 u 2 + 5 v u 2 - v 3 w u 2 + 4 v 2 w u 2 -5 v w u 2 + 2 w u 2 - u 2 -2 v 3 u + 5 v 2 u -4 v u + v 3 w u -5 v 2 w u + 4 v w u - w u + u + v 3 -2 v 2 + v + v 2 w - v w$ (db)
Jones polynomial	$q 6 -3 q 5 + 7 q 4 -11 q 3 + 17 q 2 -18 q + 19-16 q -1 + 13 q -2 -7 q -3 + 3 q -4 - q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 a -2 - z 6 + 2 a 2 z 4 -3 z 4 a -2 + z 4 a -4 - z 4 - a 4 z 2 + 3 a 2 z 2 -6 z 2 a -2 + 2 z 2 a -4 + z 2 - a 4 + 2 a 2 -5 a -2 + 2 a -4 + 2-2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 + 4 a z 9 + 8 z 9 a -1 + 4 z 9 a -3 + 6 a 2 z 8 + 12 z 8 a -2 + 5 z 8 a -4 + 13 z 8 + 5 a 3 z 7 + 2 a z 7 -11 z 7 a -1 -5 z 7 a -3 + 3 z 7 a -5 + 3 a 4 z 6 -7 a 2 z 6 -41 z 6 a -2 -14 z 6 a -4 + z 6 a -6 -36 z 6 + a 5 z 5 -6 a 3 z 5 -12 a z 5 -3 z 5 a -3 -8 z 5 a -5 -5 a 4 z 4 + 3 a 2 z 4 + 55 z 4 a -2 + 16 z 4 a -4 -3 z 4 a -6 + 44 z 4 -2 a 5 z 3 + a 3 z 3 + 10 a z 3 + 8 z 3 a -1 + 6 z 3 a -3 + 5 z 3 a -5 + 3 a 4 z 2 - a 2 z 2 -39 z 2 a -2 -13 z 2 a -4 + 2 z 2 a -6 -28 z 2 + a 5 z + a 3 z -3 a z -8 z a -1 -5 z a -3 - a 4 + 13 a -2 + 6 a -4 + 9 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 L11a456. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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