L10a93

(Knotscape image)

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Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_20,12,9,11 X_2,9,3,10 X_4,20,5,19 X_14,5,15,6 X_18,13,19,14 X_16,7,17,8 X_6,15,7,16 X_8,17,1,18
Gauss code	{1, -4, 2, -5, 6, -9, 8, -10}, {4, -1, 3, -2, 7, -6, 9, -8, 10, -7, 5, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 2 v 2 u 3 - v u 3 + 2 v 3 u 2 -5 v 2 u 2 + 3 v u 2 - u 2 - v 3 u + 3 v 2 u -5 v u + 2 u - v 2 + 2 v -1$ (db)
Jones polynomial	$q^{3/2}-3 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{10}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{8}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 7 -2 z a 7 + 2 z 5 a 5 + 7 z 3 a 5 + 6 z a 5 + a 5 z -1 - z 7 a 3 -5 z 5 a 3 -9 z 3 a 3 -7 z a 3 - a 3 z -1 + z 5 a + 3 z 3 a + z a$ (db)
Kauffman polynomial	$- z 4 a 10 + z 2 a 10 -3 z 5 a 9 + 4 z 3 a 9 - z a 9 -4 z 6 a 8 + 4 z 4 a 8 -4 z 7 a 7 + 4 z 5 a 7 - z 3 a 7 -3 z 8 a 6 + 3 z 6 a 6 -2 z 4 a 6 + z 2 a 6 - z 9 a 5 -5 z 7 a 5 + 18 z 5 a 5 -21 z 3 a 5 + 9 z a 5 - a 5 z -1 -6 z 8 a 4 + 17 z 6 a 4 -16 z 4 a 4 + 4 z 2 a 4 + a 4 - z 9 a 3 -4 z 7 a 3 + 22 z 5 a 3 -26 z 3 a 3 + 10 z a 3 - a 3 z -1 -3 z 8 a 2 + 9 z 6 a 2 -6 z 4 a 2 + z 2 a 2 -3 z 7 a + 11 z 5 a -10 z 3 a + 2 z a - z 6 + 3 z 4 - z 2$ (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 =

-3 is the signature of L10a93. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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