L11a480

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 4 - v u 4 - v 2 w u 4 + v w u 4 -3 v 2 u 3 + 4 v u 3 + 3 v 2 w u 3 -4 v w u 3 + w u 3 - u 3 + 3 v 2 u 2 -6 v u 2 -3 v 2 w u 2 + 6 v w u 2 -3 w u 2 + 3 u 2 - v 2 u + 4 v u + v 2 w u -4 v w u + 3 w u -3 u - v + v w - w + 1$ (db)
Jones polynomial	$q 3 -4 q 2 + 9 q -13 + 19 q -1 -20 q -2 + 21 q -3 -17 q -4 + 13 q -5 -7 q -6 + 3 q -7 - q -8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 2 z 8 + 2 a 4 z 6 -5 a 2 z 6 + z 6 - a 6 z 4 + 8 a 4 z 4 -10 a 2 z 4 + 3 z 4 -3 a 6 z 2 + 11 a 4 z 2 -11 a 2 z 2 + 3 z 2 -2 a 6 + 6 a 4 -7 a 2 + 3 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$2 a 4 z 10 + 2 a 2 z 10 + 5 a 5 z 9 + 12 a 3 z 9 + 7 a z 9 + 6 a 6 z 8 + 10 a 4 z 8 + 12 a 2 z 8 + 8 z 8 + 5 a 7 z 7 -2 a 5 z 7 -26 a 3 z 7 -15 a z 7 + 4 z 7 a -1 + 3 a 8 z 6 -6 a 6 z 6 -32 a 4 z 6 -47 a 2 z 6 + z 6 a -2 -23 z 6 + a 9 z 5 -6 a 7 z 5 -5 a 5 z 5 + 18 a 3 z 5 + 7 a z 5 -9 z 5 a -1 -5 a 8 z 4 + a 6 z 4 + 39 a 4 z 4 + 56 a 2 z 4 -2 z 4 a -2 + 21 z 4 -2 a 9 z 3 + a 7 z 3 + 6 a 5 z 3 + a 3 z 3 + a z 3 + 3 z 3 a -1 + 3 a 8 z 2 -23 a 4 z 2 -30 a 2 z 2 -10 z 2 + a 9 z + a 7 z -3 a 5 z -6 a 3 z -3 a z - a 8 + 7 a 4 + 9 a 2 + 4 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 L11a480. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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