L11a302

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{2 (u-1) (v-1) (u v+1)^2}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$-7 q^{9/2}+8 q^{7/2}-\frac{1}{q^{7/2}}-10 q^{5/2}+\frac{2}{q^{5/2}}+10 q^{3/2}-\frac{4}{q^{3/2}}+q^{15/2}-2 q^{13/2}+4 q^{11/2}-9 \sqrt{q}+\frac{6}{\sqrt{q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 - z 7 a -3 + a z 5 -5 z 5 a -1 -5 z 5 a -3 + z 5 a -5 + 4 a z 3 -8 z 3 a -1 -8 z 3 a -3 + 4 z 3 a -5 + 4 a z -4 z a -1 -4 z a -3 + 4 z a -5 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 a -4 -2 z 9 a -1 -4 z 9 a -3 -2 z 9 a -5 + z 8 a -2 + 2 z 8 a -4 -2 z 8 a -6 -3 z 8 -3 a z 7 + 4 z 7 a -1 + 16 z 7 a -3 + 7 z 7 a -5 -2 z 7 a -7 -2 a 2 z 6 + 2 z 6 a -2 -3 z 6 a -4 + 5 z 6 a -6 - z 6 a -8 + 9 z 6 - a 3 z 5 + 9 a z 5 + z 5 a -1 -30 z 5 a -3 -14 z 5 a -5 + 7 z 5 a -7 + 5 a 2 z 4 -7 z 4 a -2 + 2 z 4 a -4 -2 z 4 a -6 + 4 z 4 a -8 -10 z 4 + 3 a 3 z 3 -9 a z 3 -13 z 3 a -1 + 23 z 3 a -3 + 18 z 3 a -5 -6 z 3 a -7 - a 2 z 2 + 2 z 2 a -4 + z 2 a -6 -4 z 2 a -8 + 2 z 2 - a 3 z + 5 a z + 6 z a -1 -6 z a -3 -5 z a -5 + z a -7 + 1- a z -1 - a -1 z -1$ (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 L11a302. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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