L11a251

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 5 + v 2 u 5 + 3 v 3 u 4 -5 v 2 u 4 + 2 v u 4 -3 v 3 u 3 + 9 v 2 u 3 -7 v u 3 + u 3 + v 3 u 2 -7 v 2 u 2 + 9 v u 2 -3 u 2 + 2 v 2 u -5 v u + 3 u + v -1$ (db)
Jones polynomial	$-q^{13/2}+4 q^{11/2}-8 q^{9/2}+13 q^{7/2}-18 q^{5/2}+20 q^{3/2}-21 \sqrt{q}+\frac{17}{\sqrt{q}}-\frac{13}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 9 a -1 - a z 7 + 6 z 7 a -1 - z 7 a -3 -4 a z 5 + 12 z 5 a -1 -4 z 5 a -3 -4 a z 3 + 8 z 3 a -1 -4 z 3 a -3 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-3 z 10 a -2 -3 z 10 -7 a z 9 -14 z 9 a -1 -7 z 9 a -3 -7 a 2 z 8 -4 z 8 a -2 -8 z 8 a -4 -3 z 8 -4 a 3 z 7 + 18 a z 7 + 41 z 7 a -1 + 12 z 7 a -3 -7 z 7 a -5 - a 4 z 6 + 19 a 2 z 6 + 20 z 6 a -2 + 12 z 6 a -4 -4 z 6 a -6 + 24 z 6 + 10 a 3 z 5 -15 a z 5 -48 z 5 a -1 -11 z 5 a -3 + 11 z 5 a -5 - z 5 a -7 + 2 a 4 z 4 -13 a 2 z 4 -23 z 4 a -2 -4 z 4 a -4 + 6 z 4 a -6 -28 z 4 -4 a 3 z 3 + 6 a z 3 + 19 z 3 a -1 + 5 z 3 a -3 -3 z 3 a -5 + z 3 a -7 + 3 a 2 z 2 + 6 z 2 a -2 - z 2 a -6 + 8 z 2 + 1- a z -1 - a -1 z -1$ (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 L11a251. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$