L10a54

(Knotscape image)

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_18,11,19,12 X_12,6,13,5 X_4,18,5,17 X_14,7,15,8 X_16,14,17,13 X_20,15,7,16 X_6,19,1,20
Gauss code	{1, -2, 3, -6, 5, -10}, {7, -1, 2, -3, 4, -5, 8, -7, 9, -8, 6, -4, 10, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + 4 v 2 u 3 -5 v u 3 + u 3 -4 v 2 u 2 + 7 v u 2 -4 u 2 + v 2 u -5 v u + 4 u + v -1$ (db)
Jones polynomial	$q^{3/2}-4 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{13}{q^{7/2}}+\frac{12}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 7 -2 z a 7 + 2 z 5 a 5 + 6 z 3 a 5 + 4 z a 5 - z 7 a 3 -4 z 5 a 3 -5 z 3 a 3 -2 z a 3 + a 3 z -1 + z 5 a + 2 z 3 a - z a - a z -1$ (db)
Kauffman polynomial	$- z 4 a 10 + z 2 a 10 -3 z 5 a 9 + 3 z 3 a 9 - z a 9 -5 z 6 a 8 + 5 z 4 a 8 -2 z 2 a 8 -6 z 7 a 7 + 7 z 5 a 7 -4 z 3 a 7 + z a 7 -5 z 8 a 6 + 5 z 6 a 6 - z 4 a 6 -2 z 9 a 5 -6 z 7 a 5 + 21 z 5 a 5 -18 z 3 a 5 + 5 z a 5 -10 z 8 a 4 + 25 z 6 a 4 -19 z 4 a 4 + 6 z 2 a 4 -2 z 9 a 3 -4 z 7 a 3 + 23 z 5 a 3 -19 z 3 a 3 + 2 z a 3 + a 3 z -1 -5 z 8 a 2 + 14 z 6 a 2 -10 z 4 a 2 + 3 z 2 a 2 - a 2 -4 z 7 a + 12 z 5 a -8 z 3 a - z a + a z -1 - z 6 + 2 z 4$ (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 =

-3 is the signature of L10a54. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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