L9a37

(Knotscape image)

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Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_18,12,9,11 X_2,9,3,10 X_4,18,5,17 X_16,8,17,7 X_14,6,15,5 X_6,16,7,15 X_8,14,1,13
Gauss code	{1, -4, 2, -5, 7, -8, 6, -9}, {4, -1, 3, -2, 9, -7, 8, -6, 5, -3}

A Braid Representative

A Morse Link Presentation

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

3 is the signature of L9a37. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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