L11a249

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 5 + 2 v 2 u 5 - v u 5 + 3 v 3 u 4 -7 v 2 u 4 + 4 v u 4 -3 v 3 u 3 + 12 v 2 u 3 -10 v u 3 + u 3 + v 3 u 2 -10 v 2 u 2 + 12 v u 2 -3 u 2 + 4 v 2 u -7 v u + 3 u - v 2 + 2 v -1$ (db)
Jones polynomial	$-q^{13/2}+5 q^{11/2}-11 q^{9/2}+18 q^{7/2}-25 q^{5/2}+28 q^{3/2}-29 \sqrt{q}+\frac{24}{\sqrt{q}}-\frac{18}{q^{3/2}}+\frac{11}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 9 a -1 - a z 7 + 5 z 7 a -1 - z 7 a -3 -3 a z 5 + 7 z 5 a -1 -3 z 5 a -3 - a z 3 - z 3 a -3 + 2 a z -4 z a -1 + 2 z a -3 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-4 z 10 a -2 -4 z 10 -10 a z 9 -21 z 9 a -1 -11 z 9 a -3 -10 a 2 z 8 -18 z 8 a -2 -14 z 8 a -4 -14 z 8 -5 a 3 z 7 + 15 a z 7 + 37 z 7 a -1 + 6 z 7 a -3 -11 z 7 a -5 - a 4 z 6 + 21 a 2 z 6 + 48 z 6 a -2 + 18 z 6 a -4 -5 z 6 a -6 + 47 z 6 + 9 a 3 z 5 + 2 a z 5 -8 z 5 a -1 + 14 z 5 a -3 + 14 z 5 a -5 - z 5 a -7 + a 4 z 4 -11 a 2 z 4 -30 z 4 a -2 -4 z 4 a -4 + 4 z 4 a -6 -34 z 4 -3 a 3 z 3 -9 a z 3 -15 z 3 a -1 -13 z 3 a -3 -4 z 3 a -5 + a 2 z 2 + 3 z 2 a -2 - z 2 a -4 + 5 z 2 + 4 a z + 8 z a -1 + 4 z a -3 + 1- a z -1 - a -1 z -1$ (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 L11a249. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{16}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{15}$
$r = 1$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = 2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = 3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$