L11a490

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$u 5 + 3 v u 4 - v w u 4 + 2 w u 4 -4 u 4 -6 v u 3 + 4 v w u 3 -5 w u 3 + 6 u 3 + 5 v u 2 -6 v w u 2 + 6 w u 2 -4 u 2 -2 v u + 4 v w u -3 w u + u - v w$ (db)
Jones polynomial	$- q 8 + 3 q 7 -7 q 6 + 13 q 5 -18 q 4 + 21 q 3 -20 q 2 + 19 q -13 + 9 q -1 -3 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 -3 z 4 a -4 -2 z 4 + a 2 z 2 -5 z 2 a -2 -2 z 2 a -4 + 3 z 2 a -6 - z 2 + a 2 -9 a -2 + 4 a -4 + 2 a -6 - a -8 + 3-5 a -2 z -2 + 4 a -4 z -2 - a -6 z -2 + 2 z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 a -4 + 4 z 9 a -1 + 8 z 9 a -3 + 4 z 9 a -5 + 16 z 8 a -2 + 16 z 8 a -4 + 6 z 8 a -6 + 6 z 8 + 3 a z 7 + 4 z 7 a -1 + 3 z 7 a -3 + 7 z 7 a -5 + 5 z 7 a -7 + a 2 z 6 -46 z 6 a -2 -38 z 6 a -4 -5 z 6 a -6 + 3 z 6 a -8 -15 z 6 -6 a z 5 -32 z 5 a -1 -48 z 5 a -3 -29 z 5 a -5 -6 z 5 a -7 + z 5 a -9 -3 a 2 z 4 + 52 z 4 a -2 + 35 z 4 a -4 - z 4 a -6 -5 z 4 a -8 + 18 z 4 + 3 a z 3 + 39 z 3 a -1 + 68 z 3 a -3 + 37 z 3 a -5 + 3 z 3 a -7 -2 z 3 a -9 + 3 a 2 z 2 -44 z 2 a -2 -22 z 2 a -4 + 4 z 2 a -6 + 3 z 2 a -8 -18 z 2 -23 z a -1 -41 z a -3 -21 z a -5 -2 z a -7 + z a -9 - a 2 + 23 a -2 + 14 a -4 - a -8 + 10 + 5 a -1 z -1 + 9 a -3 z -1 + 5 a -5 z -1 + a -7 z -1 -5 a -2 z -2 -4 a -4 z -2 - a -6 z -2 -2 z -2$ (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 L11a490. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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