L11a408

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v 2 u 2 -5 v u 2 - v 2 w u 2 + 3 v w u 2 -2 w u 2 + 2 u 2 -5 v 2 u + 8 v u + 3 v 2 w u -8 v w u + 5 w u -3 u + 2 v 2 -3 v -2 v 2 w + 5 v w -2 w + 1$ (db)
Jones polynomial	$- q 5 + 4 q 4 -8 q 3 + 14 q 2 -17 q + 21-19 q -1 + 17 q -2 -12 q -3 + 7 q -4 -3 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a 6 -3 z 2 a 4 -2 a 4 + 3 z 4 a 2 + 4 z 2 a 2 + a 2 z -2 + 3 a 2 - z 6 -2 z 4 -5 z 2 -2 z -2 -4 + 2 z 4 a -2 + 2 z 2 a -2 + a -2 z -2 + 2 a -2 - z 2 a -4$ (db)
Kauffman polynomial	$a 2 z 10 + z 10 + 3 a 3 z 9 + 7 a z 9 + 4 z 9 a -1 + 4 a 4 z 8 + 10 a 2 z 8 + 7 z 8 a -2 + 13 z 8 + 3 a 5 z 7 + 3 a 3 z 7 + 7 z 7 a -1 + 7 z 7 a -3 + a 6 z 6 -6 a 4 z 6 -20 a 2 z 6 -5 z 6 a -2 + 4 z 6 a -4 -22 z 6 -8 a 5 z 5 -19 a 3 z 5 -20 a z 5 -20 z 5 a -1 -10 z 5 a -3 + z 5 a -5 -3 a 6 z 4 -2 a 4 z 4 + 5 a 2 z 4 -5 z 4 a -2 -6 z 4 a -4 + 5 z 4 + 7 a 5 z 3 + 18 a 3 z 3 + 14 a z 3 + 8 z 3 a -1 + 4 z 3 a -3 - z 3 a -5 + 3 a 6 z 2 + 5 a 4 z 2 + 6 a 2 z 2 + 6 z 2 a -2 + 3 z 2 a -4 + 7 z 2 -2 a 5 z -6 a 3 z + 4 z a -1 - a 6 - a 4 -3 a 2 -4 a -2 -6-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

0 is the signature of L11a408. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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