L11a137

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_2,11,3,12 X_14,3,15,4 X_12,5,13,6 X_22,13,5,14 X_4,21,1,22 X_20,16,21,15 X_16,8,17,7 X_18,10,19,9 X_8,18,9,17 X_10,20,11,19
Gauss code	{1, -2, 3, -6}, {4, -1, 8, -10, 9, -11, 2, -4, 5, -3, 7, -8, 10, -9, 11, -7, 6, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 5 + 2 u 5 + 5 v u 4 -6 u 4 -6 v u 3 + 6 u 3 + 6 v u 2 -6 u 2 -6 v u + 5 u + 2 v -2$ (db)
Jones polynomial	$q^{11/2}-4 q^{9/2}+8 q^{7/2}-12 q^{5/2}+16 q^{3/2}-17 \sqrt{q}+\frac{16}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- a z 7 - z 7 a -1 + a 3 z 5 -4 a z 5 -3 z 5 a -1 + z 5 a -3 + 3 a 3 z 3 -6 a z 3 - z 3 a -1 + 2 z 3 a -3 + 2 a 3 z -6 a z + 2 z a -1 + 2 a 3 z -1 -3 a z -1 + a -1 z -1$ (db)
Kauffman polynomial	$-2 a 2 z 10 -2 z 10 -4 a 3 z 9 -10 a z 9 -6 z 9 a -1 -3 a 4 z 8 -9 z 8 a -2 -6 z 8 - a 5 z 7 + 15 a 3 z 7 + 33 a z 7 + 7 z 7 a -1 -10 z 7 a -3 + 12 a 4 z 6 + 17 a 2 z 6 + 11 z 6 a -2 -8 z 6 a -4 + 24 z 6 + 4 a 5 z 5 -18 a 3 z 5 -39 a z 5 - z 5 a -1 + 12 z 5 a -3 -4 z 5 a -5 -13 a 4 z 4 -23 a 2 z 4 + 8 z 4 a -4 - z 4 a -6 -19 z 4 -4 a 5 z 3 + 11 a 3 z 3 + 24 a z 3 + 4 z 3 a -1 -3 z 3 a -3 + 2 z 3 a -5 + 3 a 4 z 2 + 10 a 2 z 2 -2 z 2 a -2 -2 z 2 a -4 + 7 z 2 + a 5 z -5 a 3 z -8 a z -2 z a -1 -3 a 2 - a -2 -3 + 2 a 3 z -1 + 3 a z -1 + a -1 z -1$ (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 L11a137. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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