L11a370

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 -2 v 3 u 3 + 5 v 2 u 3 -5 v u 3 + u 3 - v 4 u 2 + 5 v 3 u 2 -9 v 2 u 2 + 5 v u 2 - u 2 + v 4 u -5 v 3 u + 5 v 2 u -2 v u + v 3 - v 2$ (db)
Jones polynomial	$q^{5/2}-4 q^{3/2}+8 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{15}{q^{3/2}}-\frac{17}{q^{5/2}}+\frac{15}{q^{7/2}}-\frac{13}{q^{9/2}}+\frac{9}{q^{11/2}}-\frac{5}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 7 + 2 z a 7 + a 7 z -1 - z 5 a 5 -2 z 3 a 5 -3 z a 5 - a 5 z -1 -2 z 5 a 3 -4 z 3 a 3 -3 z a 3 - z 5 a + z a + z 3 a -1$ (db)
Kauffman polynomial	$- a 6 z 10 - a 4 z 10 -2 a 7 z 9 -6 a 5 z 9 -4 a 3 z 9 -2 a 8 z 8 -3 a 6 z 8 -9 a 4 z 8 -8 a 2 z 8 - a 9 z 7 + 4 a 7 z 7 + 11 a 5 z 7 -4 a 3 z 7 -10 a z 7 + 8 a 8 z 6 + 15 a 6 z 6 + 22 a 4 z 6 + 7 a 2 z 6 -8 z 6 + 5 a 9 z 5 + 3 a 7 z 5 + a 5 z 5 + 20 a 3 z 5 + 13 a z 5 -4 z 5 a -1 -10 a 8 z 4 -14 a 6 z 4 -9 a 4 z 4 + 4 a 2 z 4 - z 4 a -2 + 8 z 4 -8 a 9 z 3 -7 a 7 z 3 -4 a 5 z 3 -13 a 3 z 3 -6 a z 3 + 2 z 3 a -1 + 4 a 8 z 2 + 5 a 6 z 2 -3 a 2 z 2 -2 z 2 + 4 a 9 z -2 a 5 z + 3 a 3 z + a z - a 6 + a 7 z -1 + a 5 z -1$ (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 L11a370. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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