L9a31

(Knotscape image)

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

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Planar diagram presentation	X₈₁₉₂ X_16,9,17,10 X₆₇₁₈ X_18,13,7,14 X_10,4,11,3 X_14,6,15,5 X_4,12,5,11 X_12,17,13,18 X_2,16,3,15
Gauss code	{1, -9, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -8, 4, -6, 9, -2, 8, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + 2 v 2 u 3 -4 v u 3 + u 3 -2 v 2 u 2 + 5 v u 2 -2 u 2 + v 2 u -4 v u + 2 u + v -1$ (db)
Jones polynomial	$-q^{7/2}+3 q^{5/2}-6 q^{3/2}+7 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 5 a z 5 - z 5 a -1 -3 a 3 z 3 + 9 a z 3 -3 z 3 a -1 -3 a 3 z + 7 a z -3 z a -1 - a 3 z -1 + 3 a z -1 -2 a -1 z -1$ (db)
Kauffman polynomial	$-2 a 2 z 8 -2 z 8 -5 a 3 z 7 -9 a z 7 -4 z 7 a -1 -5 a 4 z 6 -4 a 2 z 6 -3 z 6 a -2 -2 z 6 -3 a 5 z 5 + 8 a 3 z 5 + 20 a z 5 + 8 z 5 a -1 - z 5 a -3 - a 6 z 4 + 7 a 4 z 4 + 12 a 2 z 4 + 6 z 4 a -2 + 10 z 4 + 3 a 5 z 3 -8 a 3 z 3 -18 a z 3 -5 z 3 a -1 + 2 z 3 a -3 + a 6 z 2 -4 a 4 z 2 -10 a 2 z 2 -2 z 2 a -2 -7 z 2 + 4 a 3 z + 9 a z + 4 z a -1 - z a -3 + a 4 + 3 a 2 + 3- a 3 z -1 -3 a z -1 -2 a -1 z -1$ (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 L9a31. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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