L11a512

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_16,5,17,6 X_22,15,13,16 X_14,4,15,3 X_4,14,5,13 X_12,17,7,18 X_10,19,11,20 X_18,9,19,10 X_20,11,21,12 X₂₇₃₈ X_6,21,1,22
Gauss code	{1, -10, 4, -5, 2, -11}, {10, -1, 8, -7, 9, -6}, {5, -4, 3, -2, 6, -8, 7, -9, 11, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 - v w 2 u 3 - v u 3 - v 2 w u 3 + 2 v w u 3 -2 v 2 u 2 - v 2 w 2 u 2 + 3 v w 2 u 2 - w 2 u 2 + 3 v u 2 + 4 v 2 w u 2 -7 v w u 2 + 2 w u 2 - u 2 + v 2 u + v 2 w 2 u -3 v w 2 u + 2 w 2 u -3 v u -2 v 2 w u + 7 v w u -4 w u + u + v w 2 - w 2 + v -2 v w + w$ (db)
Jones polynomial	$1-3 q -1 + 7 q -2 -11 q -3 + 17 q -4 -18 q -5 + 20 q -6 -17 q -7 + 13 q -8 -8 q -9 + 4 q -10 - q -11$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- z 2 a 10 - a 10 + 3 z 4 a 8 + 7 z 2 a 8 + a 8 z -2 + 4 a 8 -2 z 6 a 6 -7 z 4 a 6 -10 z 2 a 6 -2 a 6 z -2 -8 a 6 - z 6 a 4 - z 4 a 4 + 4 z 2 a 4 + a 4 z -2 + 5 a 4 + z 4 a 2 + 2 z 2 a 2$ (db)
Kauffman polynomial	$z 5 a 13 - z 3 a 13 + 4 z 6 a 12 -6 z 4 a 12 + 2 z 2 a 12 + 7 z 7 a 11 -11 z 5 a 11 + 5 z 3 a 11 - z a 11 + 7 z 8 a 10 -7 z 6 a 10 - z 4 a 10 + a 10 + 4 z 9 a 9 + 6 z 7 a 9 -23 z 5 a 9 + 18 z 3 a 9 -3 z a 9 + z 10 a 8 + 14 z 8 a 8 -36 z 6 a 8 + 39 z 4 a 8 -24 z 2 a 8 - a 8 z -2 + 8 a 8 + 8 z 9 a 7 -9 z 7 a 7 -6 z 5 a 7 + 14 z 3 a 7 -8 z a 7 + 2 a 7 z -1 + z 10 a 6 + 12 z 8 a 6 -40 z 6 a 6 + 53 z 4 a 6 -37 z 2 a 6 -2 a 6 z -2 + 12 a 6 + 4 z 9 a 5 -5 z 7 a 5 -3 z 5 a 5 + 7 z 3 a 5 -6 z a 5 + 2 a 5 z -1 + 5 z 8 a 4 -14 z 6 a 4 + 16 z 4 a 4 -13 z 2 a 4 - a 4 z -2 + 6 a 4 + 3 z 7 a 3 -8 z 5 a 3 + 5 z 3 a 3 + z 6 a 2 -3 z 4 a 2 + 2 z 2 a 2$ (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 L11a512. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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