L11a87

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 2 u 5 + 5 v u 4 -7 u 4 -10 v u 3 + 12 u 3 + 12 v u 2 -10 u 2 -7 v u + 5 u + 2 v -1$ (db)
Jones polynomial	$-q^{7/2}+5 q^{5/2}-11 q^{3/2}+17 \sqrt{q}-\frac{22}{\sqrt{q}}+\frac{24}{q^{3/2}}-\frac{24}{q^{5/2}}+\frac{19}{q^{7/2}}-\frac{14}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + 3 z 3 a 5 + 6 z a 5 + 4 a 5 z -1 -3 z 5 a 3 -8 z 3 a 3 -9 z a 3 -4 a 3 z -1 + z 7 a + 3 z 5 a + 4 z 3 a + 2 z a + a z -1 - z 5 a -1 - z 3 a -1$ (db)
Kauffman polynomial	$-2 a 4 z 10 -2 a 2 z 10 -4 a 5 z 9 -12 a 3 z 9 -8 a z 9 -4 a 6 z 8 -9 a 4 z 8 -18 a 2 z 8 -13 z 8 -3 a 7 z 7 - a 5 z 7 + 12 a 3 z 7 - a z 7 -11 z 7 a -1 - a 8 z 6 + 4 a 6 z 6 + 19 a 4 z 6 + 37 a 2 z 6 -5 z 6 a -2 + 18 z 6 + 8 a 7 z 5 + 18 a 5 z 5 + 14 a 3 z 5 + 20 a z 5 + 15 z 5 a -1 - z 5 a -3 + 3 a 8 z 4 + 7 a 6 z 4 - a 4 z 4 -13 a 2 z 4 + 4 z 4 a -2 -4 z 4 -8 a 7 z 3 -25 a 5 z 3 -25 a 3 z 3 -12 a z 3 -4 z 3 a -1 -3 a 8 z 2 -11 a 6 z 2 -15 a 4 z 2 -8 a 2 z 2 - z 2 + 4 a 7 z + 16 a 5 z + 15 a 3 z + 3 a z + a 8 + 4 a 6 + 7 a 4 + 4 a 2 + 1- a 7 z -1 -4 a 5 z -1 -4 a 3 z -1 - a z -1$ (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 L11a87. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{6}$
$r = -3$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -2$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 0$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = 1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$