L11a115

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_16,8,17,7 X_22,18,5,17 X_18,11,19,12 X_20,9,21,10 X_10,19,11,20 X_8,21,9,22 X_12,16,13,15 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -8, 6, -7, 5, -9, 11, -2, 9, -3, 4, -5, 7, -6, 8, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 4 u 3 + 7 v u 2 -10 u 2 -10 v u + 7 u + 4 v -2$ (db)
Jones polynomial	$q^{5/2}-4 q^{3/2}+7 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{13}{q^{3/2}}-\frac{15}{q^{5/2}}+\frac{14}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{5}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 9 z -1 -3 z a 7 -2 a 7 z -1 + 3 z 3 a 5 + 3 z a 5 + a 5 z -1 - z 5 a 3 + a 3 z -1 - z 5 a - z 3 a -2 z a - a z -1 + z 3 a -1$ (db)
Kauffman polynomial	$- a 6 z 10 - a 4 z 10 -2 a 7 z 9 -6 a 5 z 9 -4 a 3 z 9 -2 a 8 z 8 -3 a 6 z 8 -8 a 4 z 8 -7 a 2 z 8 - a 9 z 7 + 4 a 7 z 7 + 15 a 5 z 7 + 2 a 3 z 7 -8 a z 7 + 8 a 8 z 6 + 20 a 6 z 6 + 29 a 4 z 6 + 10 a 2 z 6 -7 z 6 + 5 a 9 z 5 + 7 a 7 z 5 -4 a 5 z 5 + 7 a 3 z 5 + 9 a z 5 -4 z 5 a -1 -9 a 8 z 4 -25 a 6 z 4 -30 a 4 z 4 -6 a 2 z 4 - z 4 a -2 + 7 z 4 -8 a 9 z 3 -18 a 7 z 3 -11 a 5 z 3 -4 a 3 z 3 + 3 z 3 a -1 + 4 a 8 z 2 + 12 a 6 z 2 + 12 a 4 z 2 + 4 a 2 z 2 + 5 a 9 z + 12 a 7 z + 8 a 5 z -2 a 3 z -3 a z - a 8 -3 a 6 -2 a 4 - a 2 - a 9 z -1 -2 a 7 z -1 - a 5 z -1 + a 3 z -1 + a z -1$ (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 L11a115. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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