L10a44

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 2 u 5 + 2 v u 4 -2 u 4 -2 v u 3 + 2 u 3 + 2 v u 2 -2 u 2 -2 v u + 2 u + 2 v -1$ (db)
Jones polynomial	$-q^{15/2}+3 q^{13/2}-4 q^{11/2}+6 q^{9/2}-7 q^{7/2}+7 q^{5/2}-6 q^{3/2}+4 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 -2 z 5 a -1 + 5 z 5 a -3 - z 5 a -5 + a z 3 -9 z 3 a -1 + 8 z 3 a -3 -3 z 3 a -5 + 4 a z -11 z a -1 + 6 z a -3 - z a -5 + 3 a z -1 -5 a -1 z -1 + 2 a -3 z -1$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -4 z 8 a -2 -3 z 8 a -4 - z 8 - a z 7 + 2 z 7 a -1 - z 7 a -3 -4 z 7 a -5 + 14 z 6 a -2 + 7 z 6 a -4 -4 z 6 a -6 + 3 z 6 + 6 a z 5 + 6 z 5 a -1 + 11 z 5 a -3 + 7 z 5 a -5 -4 z 5 a -7 -8 z 4 a -2 -4 z 4 a -4 + 3 z 4 a -6 -3 z 4 a -8 + 2 z 4 -12 a z 3 -16 z 3 a -1 -11 z 3 a -3 -3 z 3 a -5 + 3 z 3 a -7 - z 3 a -9 -8 z 2 a -2 + z 2 a -4 + 2 z 2 a -6 + 2 z 2 a -8 -9 z 2 + 10 a z + 14 z a -1 + 5 z a -3 + z a -5 + 5 a -2 - a -6 + 5-3 a z -1 -5 a -1 z -1 -2 a -3 z -1$ (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 =

3 is the signature of L10a44. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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