L11a88

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 5 u 3 + 7 v u 2 -9 u 2 -9 v u + 7 u + 5 v -2$ (db)
Jones polynomial	$q^{15/2}-3 q^{13/2}+6 q^{11/2}-9 q^{9/2}+13 q^{7/2}-15 q^{5/2}+14 q^{3/2}-13 \sqrt{q}+\frac{9}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 5 a -1 + z 5 a -3 -2 a z 3 -2 z 3 a -5 + a 3 z -2 a z - z a -3 - z a -5 + z a -7 + a 3 z -1 - a z -1 + a -1 z -1 -2 a -3 z -1 + a -5 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 a -4 -3 z 9 a -1 -7 z 9 a -3 -4 z 9 a -5 -6 z 8 a -2 -8 z 8 a -4 -5 z 8 a -6 -3 z 8 -3 a z 7 + 5 z 7 a -1 + 21 z 7 a -3 + 10 z 7 a -5 -3 z 7 a -7 -2 a 2 z 6 + 23 z 6 a -2 + 38 z 6 a -4 + 17 z 6 a -6 - z 6 a -8 + z 6 - a 3 z 5 + 3 a z 5 -12 z 5 a -1 -30 z 5 a -3 -5 z 5 a -5 + 9 z 5 a -7 + 3 a 2 z 4 -37 z 4 a -2 -58 z 4 a -4 -18 z 4 a -6 + 3 z 4 a -8 + 3 z 4 + 3 a 3 z 3 + 2 a z 3 + 16 z 3 a -1 + 21 z 3 a -3 - z 3 a -5 -5 z 3 a -7 + 23 z 2 a -2 + 34 z 2 a -4 + 10 z 2 a -6 - z 2 a -8 -3 a 3 z -3 a z -7 z a -1 -9 z a -3 - z a -5 + z a -7 - a 2 -4 a -2 -7 a -4 -3 a -6 + a 3 z -1 + a z -1 + a -1 z -1 + 2 a -3 z -1 + a -5 z -1$ (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 L11a88. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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