L11a468

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + 3 v 2 u 3 -3 v u 3 -2 v 2 w u 3 + 2 v w u 3 + u 3 -2 v 2 u 2 + 4 v u 2 + 3 v 2 w u 2 -4 v w u 2 + 2 w u 2 -3 u 2 -2 v u - v 2 w u + 3 v w u -3 w u + 2 u - v w + w$ (db)
Jones polynomial	$1-2 q -1 + 5 q -2 -8 q -3 + 12 q -4 -13 q -5 + 15 q -6 -12 q -7 + 10 q -8 -6 q -9 + 3 q -10 - q -11$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- z 2 a 10 -2 a 10 + 3 z 4 a 8 + 9 z 2 a 8 + a 8 z -2 + 6 a 8 -2 z 6 a 6 -8 z 4 a 6 -11 z 2 a 6 -2 a 6 z -2 -8 a 6 - z 6 a 4 -2 z 4 a 4 + 2 z 2 a 4 + a 4 z -2 + 3 a 4 + z 4 a 2 + 3 z 2 a 2 + a 2$ (db)
Kauffman polynomial	$z 5 a 13 -2 z 3 a 13 + z a 13 + 3 z 6 a 12 -6 z 4 a 12 + 3 z 2 a 12 - a 12 + 4 z 7 a 11 -5 z 5 a 11 -2 z 3 a 11 + z a 11 + 4 z 8 a 10 -4 z 6 a 10 -2 z 4 a 10 + z 2 a 10 + 3 z 9 a 9 -2 z 7 a 9 -4 z 5 a 9 + 8 z 3 a 9 -3 z a 9 + z 10 a 8 + 6 z 8 a 8 -25 z 6 a 8 + 42 z 4 a 8 -29 z 2 a 8 - a 8 z -2 + 10 a 8 + 6 z 9 a 7 -16 z 7 a 7 + 14 z 5 a 7 + 7 z 3 a 7 -10 z a 7 + 2 a 7 z -1 + z 10 a 6 + 5 z 8 a 6 -28 z 6 a 6 + 51 z 4 a 6 -40 z 2 a 6 -2 a 6 z -2 + 14 a 6 + 3 z 9 a 5 -8 z 7 a 5 + 6 z 5 a 5 + 2 z 3 a 5 -7 z a 5 + 2 a 5 z -1 + 3 z 8 a 4 -9 z 6 a 4 + 9 z 4 a 4 -9 z 2 a 4 - a 4 z -2 + 5 a 4 + 2 z 7 a 3 -6 z 5 a 3 + 3 z 3 a 3 + z 6 a 2 -4 z 4 a 2 + 4 z 2 a 2 - a 2$ (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 =

-4 is the signature of L11a468. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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