L11a31

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 3 u 5 + 4 v u 4 -5 u 4 -5 v u 3 + 5 u 3 + 5 v u 2 -5 u 2 -5 v u + 4 u + 3 v -1$ (db)
Jones polynomial	$q^{17/2}-3 q^{15/2}+7 q^{13/2}-10 q^{11/2}+13 q^{9/2}-15 q^{7/2}+14 q^{5/2}-12 q^{3/2}+8 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 -2 z 5 a -1 + 3 z 5 a -3 -2 z 5 a -5 + a z 3 -6 z 3 a -1 -5 z 3 a -5 + z 3 a -7 + 3 a z -2 z a -1 -5 z a -3 + 2 z a -7 + a z -1 + a -1 z -1 -4 a -3 z -1 + 2 a -5 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 a -4 -3 z 9 a -1 -7 z 9 a -3 -4 z 9 a -5 -5 z 8 a -2 -9 z 8 a -4 -6 z 8 a -6 -2 z 8 - a z 7 + 11 z 7 a -1 + 22 z 7 a -3 + 3 z 7 a -5 -7 z 7 a -7 + 30 z 6 a -2 + 35 z 6 a -4 + 6 z 6 a -6 -6 z 6 a -8 + 7 z 6 + 5 a z 5 -13 z 5 a -1 -27 z 5 a -3 + 3 z 5 a -5 + 9 z 5 a -7 -3 z 5 a -9 -42 z 4 a -2 -44 z 4 a -4 + 2 z 4 a -6 + 8 z 4 a -8 - z 4 a -10 -5 z 4 -8 a z 3 + 10 z 3 a -1 + 27 z 3 a -3 + 2 z 3 a -5 -5 z 3 a -7 + 2 z 3 a -9 + 23 z 2 a -2 + 27 z 2 a -4 -4 z 2 a -6 -6 z 2 a -8 + z 2 a -10 - z 2 + 5 a z -6 z a -1 -17 z a -3 -5 z a -5 + z a -7 -5 a -2 -6 a -4 + a -6 + 2 a -8 + 1- a z -1 + a -1 z -1 + 4 a -3 z -1 + 2 a -5 z -1$ (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 =

3 is the signature of L11a31. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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