L10a41

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_16,10,17,9 X_14,12,15,11 X_10,16,11,15 X_20,17,5,18 X_18,7,19,8 X_8,19,9,20 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, 7, -8, 3, -5, 4, -2, 10, -4, 5, -3, 6, -7, 8, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 u 3 + 5 v u 2 -5 u 2 -5 v u + 5 u + 2 v -2$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+5 q^{5/2}-7 q^{3/2}+8 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{8}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 5 - a 5 z -1 + 2 z 3 a 3 + 4 z a 3 + 3 a 3 z -1 - z 5 a -2 z 3 a -3 z a -2 a z -1 - z 5 a -1 -2 z 3 a -1 - z a -1 + z 3 a -3 + z a -3$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 -2 a 2 z 8 -3 z 8 a -2 -5 z 8 -2 a 3 z 7 -2 a z 7 -3 z 7 a -1 -3 z 7 a -3 -2 a 4 z 6 + a 2 z 6 + 8 z 6 a -2 - z 6 a -4 + 12 z 6 -2 a 5 z 5 -2 a 3 z 5 + 5 a z 5 + 15 z 5 a -1 + 10 z 5 a -3 - a 6 z 4 - a 4 z 4 -2 a 2 z 4 -3 z 4 a -2 + 3 z 4 a -4 -8 z 4 + 3 a 5 z 3 + 7 a 3 z 3 -12 z 3 a -1 -8 z 3 a -3 + 2 a 6 z 2 + 5 a 4 z 2 + 4 a 2 z 2 - z 2 a -2 -2 z 2 a -4 + 2 z 2 -2 a 5 z -7 a 3 z -5 a z + 2 z a -1 + 2 z a -3 - a 6 -3 a 4 -3 a 2 + a 5 z -1 + 3 a 3 z -1 + 2 a z -1$ (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 L10a41. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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