L11a337

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 -4 v 2 u 4 + 5 v u 4 - u 4 -2 v 3 u 3 + 9 v 2 u 3 -10 v u 3 + 2 u 3 + 2 v 3 u 2 -10 v 2 u 2 + 9 v u 2 -2 u 2 - v 3 u + 5 v 2 u -4 v u - v 2$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{13}{q^{11/2}}-\frac{19}{q^{13/2}}+\frac{22}{q^{15/2}}-\frac{22}{q^{17/2}}+\frac{19}{q^{19/2}}-\frac{15}{q^{21/2}}+\frac{9}{q^{23/2}}-\frac{4}{q^{25/2}}+\frac{1}{q^{27/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z a 13 - a 13 z -1 + 4 z 3 a 11 + 7 z a 11 + 3 a 11 z -1 -3 z 5 a 9 -7 z 3 a 9 -5 z a 9 -2 a 9 z -1 -3 z 5 a 7 -7 z 3 a 7 -4 z a 7 - z 5 a 5 -2 z 3 a 5 - z a 5$ (db)
Kauffman polynomial	$- z 6 a 16 + 2 z 4 a 16 - z 2 a 16 -4 z 7 a 15 + 9 z 5 a 15 -6 z 3 a 15 + z a 15 -7 z 8 a 14 + 15 z 6 a 14 -8 z 4 a 14 + 2 z 2 a 14 - a 14 -6 z 9 a 13 + 6 z 7 a 13 + 8 z 5 a 13 -6 z 3 a 13 + a 13 z -1 -2 z 10 a 12 -12 z 8 a 12 + 36 z 6 a 12 -31 z 4 a 12 + 15 z 2 a 12 -3 a 12 -12 z 9 a 11 + 21 z 7 a 11 -16 z 5 a 11 + 19 z 3 a 11 -14 z a 11 + 3 a 11 z -1 -2 z 10 a 10 -12 z 8 a 10 + 31 z 6 a 10 -28 z 4 a 10 + 12 z 2 a 10 -3 a 10 -6 z 9 a 9 + 5 z 7 a 9 -4 z 5 a 9 + 8 z 3 a 9 -8 z a 9 + 2 a 9 z -1 -7 z 8 a 8 + 8 z 6 a 8 -3 z 4 a 8 - z 2 a 8 -6 z 7 a 7 + 10 z 5 a 7 -9 z 3 a 7 + 4 z a 7 -3 z 6 a 6 + 4 z 4 a 6 - z 2 a 6 - z 5 a 5 + 2 z 3 a 5 - z a 5$ (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 =

-5 is the signature of L11a337. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -6$	$i = -4$
$r = -11$	${\mathbb Z}$
$r = -10$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -9$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -8$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{7}$
$r = -7$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -6$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -5$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -4$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -1$	${\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$