L11a479

(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_22,15,17,16 X_20,10,21,9 X_8,13,9,14 X_14,17,15,18 X_16,21,5,22 X_12,20,13,19 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {7, -3, 9, -5, 8, -4}, {10, -1, 3, -6, 5, -2, 11, -9, 6, -7, 4, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 4 - v w u 4 + w u 4 - u 4 + v 2 u 3 -4 v u 3 - v 2 w u 3 + 4 v w u 3 -3 w u 3 + 3 u 3 -3 v 2 u 2 + 6 v u 2 + 3 v 2 w u 2 -6 v w u 2 + 3 w u 2 -3 u 2 + 3 v 2 u -4 v u -3 v 2 w u + 4 v w u - w u + u - v 2 + v + v 2 w - v w$ (db)
Jones polynomial	$q 4 -4 q 3 + 9 q 2 -14 q + 19-20 q -1 + 21 q -2 -16 q -3 + 13 q -4 -7 q -5 + 3 q -6 - q -7$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 2 a 6 - a 6 + 2 z 4 a 4 + 3 z 2 a 4 + a 4 z -2 + 2 a 4 - z 6 a 2 - z 4 a 2 -2 a 2 z -2 - a 2 - z 6 -2 z 4 -3 z 2 + z -2 -1 + z 4 a -2 + z 2 a -2 + a -2$ (db)
Kauffman polynomial	$2 a 4 z 10 + 2 a 2 z 10 + 4 a 5 z 9 + 12 a 3 z 9 + 8 a z 9 + 3 a 6 z 8 + 6 a 4 z 8 + 16 a 2 z 8 + 13 z 8 + a 7 z 7 -10 a 5 z 7 -28 a 3 z 7 -4 a z 7 + 13 z 7 a -1 -11 a 6 z 6 -37 a 4 z 6 -54 a 2 z 6 + 9 z 6 a -2 -19 z 6 -4 a 7 z 5 + a 5 z 5 + 7 a 3 z 5 -19 a z 5 -17 z 5 a -1 + 4 z 5 a -3 + 14 a 6 z 4 + 48 a 4 z 4 + 48 a 2 z 4 -8 z 4 a -2 + z 4 a -4 + 5 z 4 + 5 a 7 z 3 + 11 a 5 z 3 + 15 a 3 z 3 + 17 a z 3 + 7 z 3 a -1 - z 3 a -3 -9 a 6 z 2 -27 a 4 z 2 -20 a 2 z 2 + 3 z 2 a -2 + z 2 -2 a 7 z -7 a 5 z -10 a 3 z -6 a z - z a -1 + 3 a 6 + 9 a 4 + 7 a 2 - a -2 + 1 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 L11a479. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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