L11a136

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-3 v u 3 + 4 u 3 + 8 v u 2 -10 u 2 -10 v u + 8 u + 4 v -3$ (db)
Jones polynomial	$q^{5/2}-4 q^{3/2}+7 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{14}{q^{3/2}}-\frac{16}{q^{5/2}}+\frac{15}{q^{7/2}}-\frac{13}{q^{9/2}}+\frac{9}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 7 + z a 7 + a 7 z -1 - z 5 a 5 -2 z 3 a 5 -4 z a 5 -2 a 5 z -1 - z 5 a 3 + z 3 a 3 + 3 z a 3 + 2 a 3 z -1 - z 5 a - z 3 a -2 z a - a z -1 + z 3 a -1$ (db)
Kauffman polynomial	$-2 a 6 z 10 -2 a 4 z 10 -4 a 7 z 9 -10 a 5 z 9 -6 a 3 z 9 -3 a 8 z 8 -5 a 4 z 8 -8 a 2 z 8 - a 9 z 7 + 15 a 7 z 7 + 36 a 5 z 7 + 12 a 3 z 7 -8 a z 7 + 12 a 8 z 6 + 20 a 6 z 6 + 28 a 4 z 6 + 13 a 2 z 6 -7 z 6 + 4 a 9 z 5 -16 a 7 z 5 -43 a 5 z 5 -11 a 3 z 5 + 8 a z 5 -4 z 5 a -1 -13 a 8 z 4 -27 a 6 z 4 -29 a 4 z 4 -7 a 2 z 4 - z 4 a -2 + 7 z 4 -4 a 9 z 3 + 8 a 7 z 3 + 24 a 5 z 3 + 11 a 3 z 3 + 2 a z 3 + 3 z 3 a -1 + 5 a 8 z 2 + 9 a 6 z 2 + 6 a 4 z 2 + 2 a 2 z 2 + a 9 z -3 a 7 z -9 a 5 z -8 a 3 z -3 a z + a 4 + a 7 z -1 + 2 a 5 z -1 + 2 a 3 z -1 + a z -1$ (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 L11a136. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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