L11a356

(Knotscape image)

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

Planar diagram presentation	X_12,1,13,2 X_16,5,17,6 X_14,3,15,4 X_20,10,21,9 X_18,8,19,7 X_22,17,11,18 X_8,20,9,19 X_4,15,5,16 X_6,22,7,21 X_2,11,3,12 X_10,13,1,14
Gauss code	{1, -10, 3, -8, 2, -9, 5, -7, 4, -11}, {10, -1, 11, -3, 8, -2, 6, -5, 7, -4, 9, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 4 + v 2 u 4 - v 4 u 3 + 3 v 3 u 3 -4 v 2 u 3 + 2 v u 3 + v 4 u 2 -4 v 3 u 2 + 7 v 2 u 2 -4 v u 2 + u 2 + 2 v 3 u -4 v 2 u + 3 v u - u + v 2 - v$ (db)
Jones polynomial	$-q^{7/2}+3 q^{5/2}-6 q^{3/2}+9 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{13}{q^{3/2}}-\frac{13}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 3 z 7 + a z 7 - a 5 z 5 + 5 a 3 z 5 + 4 a z 5 - z 5 a -1 -4 a 5 z 3 + 9 a 3 z 3 + 4 a z 3 -3 z 3 a -1 -4 a 5 z + 6 a 3 z - a z -2 z a -1 + a 3 z -1 - a z -1$ (db)
Kauffman polynomial	$- a 4 z 10 - a 2 z 10 -2 a 5 z 9 -5 a 3 z 9 -3 a z 9 -2 a 6 z 8 -3 a 2 z 8 -5 z 8 -2 a 7 z 7 + 5 a 5 z 7 + 16 a 3 z 7 + 4 a z 7 -5 z 7 a -1 - a 8 z 6 + 4 a 6 z 6 + a 4 z 6 + 11 a 2 z 6 -3 z 6 a -2 + 12 z 6 + 7 a 7 z 5 -9 a 5 z 5 -30 a 3 z 5 - a z 5 + 12 z 5 a -1 - z 5 a -3 + 4 a 8 z 4 -2 a 4 z 4 -16 a 2 z 4 + 6 z 4 a -2 -12 z 4 -7 a 7 z 3 + 12 a 5 z 3 + 27 a 3 z 3 -3 a z 3 -9 z 3 a -1 + 2 z 3 a -3 -4 a 8 z 2 - a 6 z 2 + 4 a 4 z 2 + 6 a 2 z 2 - z 2 a -2 + 4 z 2 + 2 a 7 z -3 a 5 z -10 a 3 z -2 a z + 3 z a -1 - a 2 + a 3 z -1 + a z -1$ (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 =

-1 is the signature of L11a356. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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