L11a371

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 - v 3 u 3 + 3 v 2 u 3 -2 v u 3 + u 3 - v 4 u 2 + 3 v 3 u 2 -3 v 2 u 2 + 3 v u 2 - u 2 + v 4 u -2 v 3 u + 3 v 2 u - v u + v 3 - v 2$ (db)
Jones polynomial	$q^{21/2}-2 q^{19/2}+3 q^{17/2}-6 q^{15/2}+7 q^{13/2}-9 q^{11/2}+9 q^{9/2}-8 q^{7/2}+6 q^{5/2}-4 q^{3/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$- z 5 a -3 - z 5 a -5 - z 5 a -7 + z 3 a -1 -2 z 3 a -3 - z 3 a -5 -3 z 3 a -7 + z 3 a -9 + 2 z a -1 + 2 z a -5 -3 z a -7 + 2 z a -9 + a -5 z -1 - a -7 z -1$ (db)
Kauffman polynomial	$- z 10 a -6 - z 10 a -8 -2 z 9 a -5 -4 z 9 a -7 -2 z 9 a -9 -2 z 8 a -4 + 2 z 8 a -6 + 2 z 8 a -8 -2 z 8 a -10 -2 z 7 a -3 + 7 z 7 a -5 + 18 z 7 a -7 + 7 z 7 a -9 -2 z 7 a -11 -2 z 6 a -2 + 4 z 6 a -4 + z 6 a -8 + 6 z 6 a -10 - z 6 a -12 - z 5 a -1 + 3 z 5 a -3 -14 z 5 a -5 -33 z 5 a -7 -7 z 5 a -9 + 8 z 5 a -11 + 5 z 4 a -2 -4 z 4 a -4 -8 z 4 a -6 -5 z 4 a -8 -2 z 4 a -10 + 4 z 4 a -12 + 3 z 3 a -1 + z 3 a -3 + 12 z 3 a -5 + 23 z 3 a -7 + z 3 a -9 -8 z 3 a -11 -2 z 2 a -2 + 3 z 2 a -4 + 5 z 2 a -6 + 2 z 2 a -8 - z 2 a -10 -3 z 2 a -12 -2 z a -1 -6 z a -5 -8 z a -7 + 3 z a -9 + 3 z a -11 - a -6 + a -5 z -1 + a -7 z -1$ (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 L11a371. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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