L9a51

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_8,12,9,11 X_18,8,11,7 X_16,13,17,14 X_14,6,15,5 X_10,16,5,15 X_2,9,3,10 X_4,18,1,17
Gauss code	{1, -8, 2, -9}, {6, -1, 4, -3, 8, -7}, {3, -2, 5, -6, 7, -5, 9, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 2 -2 v u 2 - v 2 w u 2 + 2 v w u 2 - w u 2 -2 v 2 u + 4 v u + 2 v 2 w u -4 v w u + 2 w u -2 u + v 2 -2 v + 2 v w - w + 1$ (db)
Jones polynomial	$q 7 -3 q 6 + 7 q 5 -8 q 4 + 11 q 3 -10 q 2 + 9 q -6 + 4 q -1 - q -2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + 3 z 4 a -2 -2 z 4 a -4 - z 4 + 3 z 2 a -2 -4 z 2 a -4 + z 2 a -6 - z 2 + a -2 -3 a -4 + a -6 + 1 + a -2 z -2 -2 a -4 z -2 + a -6 z -2$ (db)
Kauffman polynomial	$2 z 8 a -2 + 2 z 8 a -4 + 5 z 7 a -1 + 10 z 7 a -3 + 5 z 7 a -5 + 6 z 6 a -2 + 8 z 6 a -4 + 6 z 6 a -6 + 4 z 6 + a z 5 -9 z 5 a -1 -16 z 5 a -3 -3 z 5 a -5 + 3 z 5 a -7 -18 z 4 a -2 -21 z 4 a -4 -10 z 4 a -6 + z 4 a -8 -8 z 4 - a z 3 + 3 z 3 a -1 + 2 z 3 a -3 -4 z 3 a -5 -2 z 3 a -7 + 9 z 2 a -2 + 17 z 2 a -4 + 10 z 2 a -6 - z 2 a -8 + 3 z 2 + 6 z a -3 + 6 z a -5 -3 a -2 -8 a -4 -5 a -6 + 1-2 a -3 z -1 -2 a -5 z -1 + a -2 z -2 + 2 a -4 z -2 + a -6 z -2$ (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 =

2 is the signature of L9a51. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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