L10a8

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 3 v u 4 -3 u 4 -6 v u 3 + 6 u 3 + 6 v u 2 -6 u 2 -3 v u + 3 u + v -1$ (db)
Jones polynomial	$q^{11/2}-3 q^{9/2}+7 q^{7/2}-11 q^{5/2}+12 q^{3/2}-14 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -5 z 5 a -1 + z 5 a -3 - a 3 z 3 + 7 a z 3 -11 z 3 a -1 + 3 z 3 a -3 -2 a 3 z + 9 a z -11 z a -1 + 4 z a -3 - a 3 z -1 + 4 a z -1 -4 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 -3 a 2 z 8 -5 z 8 a -2 -8 z 8 -3 a 3 z 7 -10 a z 7 -15 z 7 a -1 -8 z 7 a -3 - a 4 z 6 + 3 a 2 z 6 -6 z 6 a -4 + 10 z 6 + 9 a 3 z 5 + 35 a z 5 + 43 z 5 a -1 + 14 z 5 a -3 -3 z 5 a -5 + 3 a 4 z 4 + 9 a 2 z 4 + 14 z 4 a -2 + 7 z 4 a -4 - z 4 a -6 + 12 z 4 -9 a 3 z 3 -35 a z 3 -43 z 3 a -1 -15 z 3 a -3 + 2 z 3 a -5 -3 a 4 z 2 -12 a 2 z 2 -15 z 2 a -2 -5 z 2 a -4 + z 2 a -6 -18 z 2 + 4 a 3 z + 17 a z + 21 z a -1 + 8 z a -3 + a 4 + 4 a 2 + 4 a -2 + a -4 + 7- a 3 z -1 -4 a z -1 -4 a -1 z -1 - a -3 z -1$ (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 L10a8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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