L11a54

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_14,6,15,5 X_10,4,11,3 X_16,12,17,11 X_12,16,13,15 X_22,14,5,13 X_18,9,19,10 X_2,18,3,17 X_8,19,9,20
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -11, 9, -5, 6, -7, 8, -4, 7, -6, 10, -9, 11, -2, 3, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-4 v u 3 + 4 u 3 + 10 v u 2 -10 u 2 -10 v u + 10 u + 4 v -4$ (db)
Jones polynomial	$-q^{13/2}+4 q^{11/2}-8 q^{9/2}+12 q^{7/2}-16 q^{5/2}+18 q^{3/2}-18 \sqrt{q}+\frac{14}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a z 5 + 2 z 5 a -1 + z 5 a -3 - a 3 z 3 + a z 3 + 3 z 3 a -1 - z 3 a -5 - a 3 z + a z + z a -1 - z a -3 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 -4 a z 9 -10 z 9 a -1 -6 z 9 a -3 -4 a 2 z 8 -6 z 8 a -2 -8 z 8 a -4 -2 z 8 -3 a 3 z 7 + 8 a z 7 + 29 z 7 a -1 + 11 z 7 a -3 -7 z 7 a -5 - a 4 z 6 + 9 a 2 z 6 + 24 z 6 a -2 + 16 z 6 a -4 -4 z 6 a -6 + 14 z 6 + 9 a 3 z 5 -4 a z 5 -37 z 5 a -1 -11 z 5 a -3 + 12 z 5 a -5 - z 5 a -7 + 3 a 4 z 4 -3 a 2 z 4 -29 z 4 a -2 -12 z 4 a -4 + 6 z 4 a -6 -17 z 4 -7 a 3 z 3 + a z 3 + 18 z 3 a -1 + 6 z 3 a -3 -3 z 3 a -5 + z 3 a -7 -2 a 4 z 2 + 8 z 2 a -2 + 4 z 2 a -4 + 6 z 2 + 2 a 3 z + 2 a z -2 z a -1 -2 z a -3 + 1- a z -1 - a -1 z -1$ (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 L11a54. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 0$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$