L11n455

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,5,15,6 X_12,4,13,3 X_2,9,3,10 X_7,19,8,18 X_17,9,18,8 X_10,13,5,14 X_19,22,20,17 X_21,11,22,16 X_11,21,12,20 X_4,16,1,15
Gauss code	{1, -4, 3, -11}, {2, -1, -5, 6, 4, -7}, {-10, -3, 7, -2, 11, 9}, {-6, 5, -8, 10, -9, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 2 - v 2 w u 2 + 2 v w u 2 - w u 2 - v w x u 2 + w x u 2 -2 v 2 u + v u - v w u + w u + v 2 x u - v x u + v w x u -2 w x u + v 2 - v - v 2 x + 2 v x + w x - x$ (db)
Jones polynomial	$-q^{11/2}+2 q^{9/2}-6 q^{7/2}+5 q^{5/2}-9 q^{3/2}+7 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{1}{q^{7/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-2 z 5 a -1 + 3 a z 3 -8 z 3 a -1 + 3 z 3 a -3 - a 3 z + 6 a z -12 z a -1 + 8 z a -3 - z a -5 + 3 a z -1 -8 a -1 z -1 + 7 a -3 z -1 -2 a -5 z -1 + a z -3 -3 a -1 z -3 + 3 a -3 z -3 - a -5 z -3$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -6 z 8 a -2 -2 z 8 a -4 -4 z 8 -5 a z 7 -7 z 7 a -1 -3 z 7 a -3 - z 7 a -5 -2 a 2 z 6 + 19 z 6 a -2 + 7 z 6 a -4 + 10 z 6 + 17 a z 5 + 39 z 5 a -1 + 27 z 5 a -3 + 5 z 5 a -5 + 2 a 2 z 4 -6 z 4 a -2 -3 z 4 a -4 - z 4 -4 a 3 z 3 -24 a z 3 -52 z 3 a -1 -42 z 3 a -3 -10 z 3 a -5 - a 4 z 2 -2 a 2 z 2 -18 z 2 a -2 -9 z 2 a -4 -10 z 2 + 2 a 3 z + 16 a z + 31 z a -1 + 27 z a -3 + 10 z a -5 + 19 a -2 + 10 a -4 + 10-5 a z -1 -12 a -1 z -1 -12 a -3 z -1 -5 a -5 z -1 -6 a -2 z -2 -3 a -4 z -2 -3 z -2 + a z -3 + 3 a -1 z -3 + 3 a -3 z -3 + a -5 z -3$ (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 =

-1 is the signature of L11n455. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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