L11a396

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$3 v u 3 -2 v w u 3 + 2 w u 3 -2 u 3 -4 v u 2 + 3 v w u 2 -4 w u 2 + 3 u 2 + 4 v u -3 v w u + 4 w u -3 u -2 v + 2 v w -3 w + 2$ (db)
Jones polynomial	$- q 8 + 4 q 7 -8 q 6 + 11 q 5 -14 q 4 + 15 q 3 -14 q 2 + 12 q -6 + 5 q -1 - q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + z 6 a -4 + 3 z 4 a -2 + 2 z 4 a -4 - z 4 a -6 -2 z 4 + a 2 z 2 + 6 z 2 a -2 - z 2 a -6 -7 z 2 + 3 a 2 + 8 a -2 -2 a -4 -9 + 2 a 2 z -2 + 4 a -2 z -2 - a -4 z -2 -5 z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 a -4 + z 9 a -1 + 5 z 9 a -3 + 4 z 9 a -5 - z 8 a -2 + 5 z 8 a -4 + 7 z 8 a -6 + z 8 + a z 7 + z 7 a -1 -12 z 7 a -3 -5 z 7 a -5 + 7 z 7 a -7 + a 2 z 6 + 8 z 6 a -2 -11 z 6 a -4 -14 z 6 a -6 + 4 z 6 a -8 + 2 z 6 - a z 5 + 20 z 5 a -3 + 5 z 5 a -5 -13 z 5 a -7 + z 5 a -9 -5 a 2 z 4 -17 z 4 a -2 + 8 z 4 a -4 + 10 z 4 a -6 -6 z 4 a -8 -14 z 4 -6 a z 3 -15 z 3 a -1 -19 z 3 a -3 -5 z 3 a -5 + 4 z 3 a -7 - z 3 a -9 + 9 a 2 z 2 + 15 z 2 a -2 + z 2 a -4 -3 z 2 a -6 + 20 z 2 + 11 a z + 21 z a -1 + 13 z a -3 + 3 z a -5 -7 a 2 -10 a -2 -2 a -4 -14-5 a z -1 -9 a -1 z -1 -5 a -3 z -1 - a -5 z -1 + 2 a 2 z -2 + 4 a -2 z -2 + a -4 z -2 + 5 z -2$ (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 L11a396. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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