L10a31

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 u 3 + 5 v u 2 -5 u 2 -5 v u + 5 u + 2 v -2$ (db)
Jones polynomial	$-q^{9/2}+3 q^{7/2}-6 q^{5/2}+7 q^{3/2}-9 \sqrt{q}+\frac{9}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z a 5 + a 5 z -1 -2 z 3 a 3 -4 z a 3 -2 a 3 z -1 + z 5 a + 2 z 3 a + 2 z a + a z -1 + z 5 a -1 + 2 z 3 a -1 + 2 z a -1 + a -1 z -1 - z 3 a -3 - z a -3 - a -3 z -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -2 a 4 z 8 -6 a 2 z 8 -4 z 8 - a 5 z 7 - a 3 z 7 -6 a z 7 -6 z 7 a -1 + 9 a 4 z 6 + 22 a 2 z 6 -7 z 6 a -2 + 6 z 6 + 5 a 5 z 5 + 18 a 3 z 5 + 27 a z 5 + 8 z 5 a -1 -6 z 5 a -3 -12 a 4 z 4 -24 a 2 z 4 + 9 z 4 a -2 -3 z 4 a -4 -8 a 5 z 3 -30 a 3 z 3 -27 a z 3 + 2 z 3 a -1 + 6 z 3 a -3 - z 3 a -5 + 5 a 4 z 2 + 11 a 2 z 2 -2 z 2 a -2 + 4 z 2 + 5 a 5 z + 15 a 3 z + 10 a z -3 z a -1 -3 z a -3 - a 4 -3 a 2 - a -2 -2- a 5 z -1 -2 a 3 z -1 - a z -1 + a -1 z -1 + a -3 z -1$ (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 L10a31. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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