L11a301

(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_20,14,21,13 X_22,16,9,15 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, -6, 8, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 5 + v 2 u 5 + v 3 u 4 -3 v 2 u 4 + 2 v u 4 - v 3 u 3 + 3 v 2 u 3 -3 v u 3 + u 3 + v 3 u 2 -3 v 2 u 2 + 3 v u 2 - u 2 + 2 v 2 u -3 v u + u + v -1$ (db)
Jones polynomial	$q^{17/2}-2 q^{15/2}+4 q^{13/2}-7 q^{11/2}+9 q^{9/2}-10 q^{7/2}+9 q^{5/2}-9 q^{3/2}+6 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$- z 9 a -3 + z 7 a -1 -8 z 7 a -3 + z 7 a -5 + 6 z 5 a -1 -24 z 5 a -3 + 6 z 5 a -5 + 12 z 3 a -1 -32 z 3 a -3 + 12 z 3 a -5 + 9 z a -1 -17 z a -3 + 8 z a -5 + 2 a -1 z -1 -3 a -3 z -1 + a -5 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 a -4 -2 z 9 a -1 -5 z 9 a -3 -3 z 9 a -5 + z 8 a -2 - z 8 a -4 -4 z 8 a -6 -2 z 8 - a z 7 + 8 z 7 a -1 + 23 z 7 a -3 + 10 z 7 a -5 -4 z 7 a -7 + 6 z 6 a -2 + 12 z 6 a -4 + 12 z 6 a -6 -3 z 6 a -8 + 9 z 6 + 5 a z 5 -11 z 5 a -1 -46 z 5 a -3 -18 z 5 a -5 + 10 z 5 a -7 -2 z 5 a -9 -10 z 4 a -2 -22 z 4 a -4 -17 z 4 a -6 + 5 z 4 a -8 - z 4 a -10 -11 z 4 -7 a z 3 + 12 z 3 a -1 + 51 z 3 a -3 + 17 z 3 a -5 -12 z 3 a -7 + 3 z 3 a -9 + 9 z 2 a -2 + 17 z 2 a -4 + 6 z 2 a -6 -3 z 2 a -8 + 2 z 2 a -10 + 3 z 2 + 2 a z -9 z a -1 -21 z a -3 -8 z a -5 + 2 z a -7 -3 a -2 -3 a -4 - a -6 + 2 a -1 z -1 + 3 a -3 z -1 + a -5 z -1$ (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 =

3 is the signature of L11a301. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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