L11a80

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 2 u 5 + 4 v u 4 -8 u 4 -9 v u 3 + 11 u 3 + 11 v u 2 -9 u 2 -8 v u + 4 u + 2 v -1$ (db)
Jones polynomial	$q^{11/2}-4 q^{9/2}+10 q^{7/2}-16 q^{5/2}+20 q^{3/2}-23 \sqrt{q}+\frac{22}{\sqrt{q}}-\frac{19}{q^{3/2}}+\frac{13}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 3 a z 5 -4 z 5 a -1 + z 5 a -3 -3 a 3 z 3 + 8 a z 3 -9 z 3 a -1 + 2 z 3 a -3 + a 5 z -5 a 3 z + 8 a z -9 z a -1 + 3 z a -3 + a 5 z -1 -2 a 3 z -1 + 3 a z -1 -3 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$-2 a 2 z 10 -2 z 10 -4 a 3 z 9 -13 a z 9 -9 z 9 a -1 -3 a 4 z 8 -8 a 2 z 8 -16 z 8 a -2 -21 z 8 - a 5 z 7 + 8 a 3 z 7 + 25 a z 7 -16 z 7 a -3 + 10 a 4 z 6 + 39 a 2 z 6 + 24 z 6 a -2 -10 z 6 a -4 + 63 z 6 + 4 a 5 z 5 + 4 a 3 z 5 + 6 a z 5 + 33 z 5 a -1 + 23 z 5 a -3 -4 z 5 a -5 -11 a 4 z 4 -42 a 2 z 4 -6 z 4 a -2 + 8 z 4 a -4 - z 4 a -6 -46 z 4 -6 a 5 z 3 -16 a 3 z 3 -27 a z 3 -30 z 3 a -1 -13 z 3 a -3 + 4 a 4 z 2 + 15 a 2 z 2 -4 z 2 a -2 -4 z 2 a -4 + 11 z 2 + 4 a 5 z + 9 a 3 z + 15 a z + 14 z a -1 + 4 z a -3 -2 a 2 + 2 a -2 + a -4 - a 5 z -1 -2 a 3 z -1 -3 a z -1 -3 a -1 z -1 - a -3 z -1$ (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 L11a80. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$