L11a32

(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_12,22,13,21 X_20,14,21,13 X_14,20,15,19 X_8,16,9,15 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 5, -6, 7, -8, 9, -3, 4, -5, 8, -7, 6, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 6 u 3 + 9 v u 2 -12 u 2 -12 v u + 9 u + 6 v -2$ (db)
Jones polynomial	$q^{15/2}-3 q^{13/2}+7 q^{11/2}-11 q^{9/2}+16 q^{7/2}-19 q^{5/2}+18 q^{3/2}-17 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 5 a -1 + z 5 a -3 -2 a z 3 -2 z 3 a -1 - z 3 a -3 -2 z 3 a -5 + a 3 z + a z -4 z a -1 - z a -3 + z a -7 + 2 a z -1 -2 a -1 z -1 - a -3 z -1 + a -5 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 a -4 -4 z 9 a -1 -8 z 9 a -3 -4 z 9 a -5 -13 z 8 a -2 -12 z 8 a -4 -5 z 8 a -6 -6 z 8 -6 a z 7 -3 z 7 a -1 + 11 z 7 a -3 + 5 z 7 a -5 -3 z 7 a -7 -3 a 2 z 6 + 35 z 6 a -2 + 42 z 6 a -4 + 14 z 6 a -6 - z 6 a -8 + 5 z 6 - a 3 z 5 + 11 a z 5 + 17 z 5 a -1 + z 5 a -3 + 4 z 5 a -5 + 8 z 5 a -7 + 4 a 2 z 4 -38 z 4 a -2 -56 z 4 a -4 -15 z 4 a -6 + 3 z 4 a -8 + 4 z 4 + 2 a 3 z 3 -11 a z 3 -19 z 3 a -1 -7 z 3 a -3 -6 z 3 a -5 -5 z 3 a -7 - a 2 z 2 + 19 z 2 a -2 + 36 z 2 a -4 + 11 z 2 a -6 -2 z 2 a -8 -5 z 2 - a 3 z + 8 a z + 10 z a -1 + z a -3 + z a -5 + z a -7 -4 a -2 -9 a -4 -4 a -6 + 2-2 a z -1 -2 a -1 z -1 + a -3 z -1 + a -5 z -1$ (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 =

1 is the signature of L11a32. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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