L9a43

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 - w u 3 + u 3 + 3 v u 2 -2 v w u 2 + 3 w u 2 -2 u 2 -3 v u + 2 v w u -3 w u + 2 u + v - v w + w$ (db)
Jones polynomial	$q -2 -3 q -3 + 7 q -4 -7 q -5 + 10 q -6 -8 q -7 + 8 q -8 -5 q -9 + 2 q -10 - q -11$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 12 z -2 + 4 a 10 z -2 + 4 a 10 -6 z 2 a 8 -5 a 8 z -2 -11 a 8 + 3 z 4 a 6 + 8 z 2 a 6 + 2 a 6 z -2 + 7 a 6 + z 4 a 4 + z 2 a 4$ (db)
Kauffman polynomial	$z 5 a 13 -3 z 3 a 13 + 3 z a 13 - a 13 z -1 + 2 z 6 a 12 -4 z 4 a 12 + 3 z 2 a 12 + a 12 z -2 -2 a 12 + 2 z 7 a 11 + z 5 a 11 -11 z 3 a 11 + 13 z a 11 -5 a 11 z -1 + z 8 a 10 + 6 z 6 a 10 -16 z 4 a 10 + 14 z 2 a 10 + 4 a 10 z -2 -10 a 10 + 6 z 7 a 9 -3 z 5 a 9 -16 z 3 a 9 + 21 z a 9 -9 a 9 z -1 + z 8 a 8 + 10 z 6 a 8 -24 z 4 a 8 + 23 z 2 a 8 + 5 a 8 z -2 -14 a 8 + 4 z 7 a 7 -10 z 3 a 7 + 11 z a 7 -5 a 7 z -1 + 6 z 6 a 6 -11 z 4 a 6 + 11 z 2 a 6 + 2 a 6 z -2 -7 a 6 + 3 z 5 a 5 -2 z 3 a 5 + z 4 a 4 - z 2 a 4$ (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 =

-4 is the signature of L9a43. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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