L10a92

(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_18,14,19,13 X_14,5,15,6 X_4,17,5,18 X_16,7,17,8 X_6,15,7,16 X_8,20,1,19
Gauss code	{1, -4, 2, -7, 6, -9, 8, -10}, {4, -1, 3, -2, 5, -6, 9, -8, 7, -5, 10, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 2 v 2 u 3 - v u 3 + 2 v 3 u 2 -5 v 2 u 2 + 5 v u 2 - u 2 - v 3 u + 5 v 2 u -5 v u + 2 u - v 2 + 2 v -1$ (db)
Jones polynomial	$-q^{7/2}+3 q^{5/2}-5 q^{3/2}+8 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 -2 a 3 z 5 + 5 a z 5 - z 5 a -1 + a 5 z 3 -7 a 3 z 3 + 9 a z 3 -3 z 3 a -1 + 2 a 5 z -7 a 3 z + 5 a z -2 z a -1 + a 5 z -1 - a 3 z -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -3 a 4 z 8 -6 a 2 z 8 -3 z 8 -4 a 5 z 7 -6 a 3 z 7 -6 a z 7 -4 z 7 a -1 -3 a 6 z 6 + 2 a 4 z 6 + 11 a 2 z 6 -3 z 6 a -2 + 3 z 6 - a 7 z 5 + 7 a 5 z 5 + 17 a 3 z 5 + 19 a z 5 + 9 z 5 a -1 - z 5 a -3 + 6 a 6 z 4 + 3 a 4 z 4 -8 a 2 z 4 + 7 z 4 a -2 + 2 z 4 + 2 a 7 z 3 -3 a 5 z 3 -18 a 3 z 3 -22 a z 3 -7 z 3 a -1 + 2 z 3 a -3 -2 a 6 z 2 -2 a 4 z 2 -3 z 2 a -2 -3 z 2 - a 7 z + 3 a 5 z + 8 a 3 z + 7 a z + 3 z a -1 + a 4 - a 5 z -1 - a 3 z -1$ (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 =

-1 is the signature of L10a92. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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