L11a59

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 2 u 5 + 4 v u 4 -6 u 4 -7 v u 3 + 7 u 3 + 7 v u 2 -7 u 2 -6 v u + 4 u + 2 v -1$ (db)
Jones polynomial	$q^{11/2}-4 q^{9/2}+8 q^{7/2}-13 q^{5/2}+16 q^{3/2}-17 \sqrt{q}+\frac{17}{\sqrt{q}}-\frac{14}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 3 a z 5 -4 z 5 a -1 + z 5 a -3 -3 a 3 z 3 + 10 a z 3 -7 z 3 a -1 + 2 z 3 a -3 + a 5 z -8 a 3 z + 10 a z -6 z a -1 + z a -3 + 2 a 5 z -1 -4 a 3 z -1 + 3 a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a 2 z 10 - z 10 -2 a 3 z 9 -7 a z 9 -5 z 9 a -1 -2 a 4 z 8 -5 a 2 z 8 -10 z 8 a -2 -13 z 8 - a 5 z 7 + 2 a 3 z 7 + 12 a z 7 -2 z 7 a -1 -11 z 7 a -3 + 7 a 4 z 6 + 22 a 2 z 6 + 15 z 6 a -2 -8 z 6 a -4 + 38 z 6 + 5 a 5 z 5 + 12 a 3 z 5 + 9 a z 5 + 21 z 5 a -1 + 15 z 5 a -3 -4 z 5 a -5 -6 a 4 z 4 -19 a 2 z 4 -7 z 4 a -2 + 7 z 4 a -4 - z 4 a -6 -28 z 4 -9 a 5 z 3 -23 a 3 z 3 -22 a z 3 -16 z 3 a -1 -6 z 3 a -3 + 2 z 3 a -5 - a 4 z 2 + a 2 z 2 + z 2 a -2 - z 2 a -4 + 4 z 2 + 7 a 5 z + 15 a 3 z + 12 a z + 4 z a -1 + 2 a 4 + 3 a 2 + a -2 + 3-2 a 5 z -1 -4 a 3 z -1 -3 a z -1 - a -1 z -1$ (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 L11a59. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = -3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$