L11a248

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 3 v 2 u 3 -3 v u 3 + u 3 + 3 v 3 u 2 -11 v 2 u 2 + 11 v u 2 -3 u 2 -3 v 3 u + 11 v 2 u -11 v u + 3 u + v 3 -3 v 2 + 3 v -1$ (db)
Jones polynomial	$-q^{13/2}+5 q^{11/2}-10 q^{9/2}+15 q^{7/2}-21 q^{5/2}+23 q^{3/2}-23 \sqrt{q}+\frac{19}{\sqrt{q}}-\frac{14}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -4 z 5 a -1 + 2 z 5 a -3 - a 3 z 3 + 5 a z 3 -9 z 3 a -1 + 4 z 3 a -3 - z 3 a -5 - a 3 z + 5 a z -7 z a -1 + 3 z a -3 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 -4 a z 9 -9 z 9 a -1 -5 z 9 a -3 -6 a 2 z 8 -21 z 8 a -2 -10 z 8 a -4 -17 z 8 -4 a 3 z 7 -5 a z 7 -5 z 7 a -1 -14 z 7 a -3 -10 z 7 a -5 - a 4 z 6 + 12 a 2 z 6 + 41 z 6 a -2 + 9 z 6 a -4 -5 z 6 a -6 + 40 z 6 + 10 a 3 z 5 + 30 a z 5 + 47 z 5 a -1 + 43 z 5 a -3 + 15 z 5 a -5 - z 5 a -7 + 2 a 4 z 4 -6 a 2 z 4 -19 z 4 a -2 + 4 z 4 a -4 + 5 z 4 a -6 -26 z 4 -8 a 3 z 3 -30 a z 3 -46 z 3 a -1 -30 z 3 a -3 -6 z 3 a -5 - a 4 z 2 -3 z 2 a -4 + 4 z 2 + 2 a 3 z + 10 a z + 14 z a -1 + 6 z a -3 + 1- a z -1 - a -1 z -1$ (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 L11a248. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 0$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{12}$
$r = 1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$