L10a10

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 u 3 + 6 v u 2 -6 u 2 -6 v u + 6 u + 2 v -2$ (db)
Jones polynomial	$-q^{13/2}+3 q^{11/2}-6 q^{9/2}+8 q^{7/2}-10 q^{5/2}+11 q^{3/2}-10 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 5 a -1 + z 5 a -3 -2 a z 3 + z 3 a -1 + 2 z 3 a -3 - z 3 a -5 + a 3 z -3 a z + 3 z a -3 - z a -5 + a 3 z -1 - a z -1 - a -1 z -1 + 2 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -6 z 8 a -2 -4 z 8 a -4 -2 z 8 -2 a z 7 -3 z 7 a -1 -6 z 7 a -3 -5 z 7 a -5 -2 a 2 z 6 + 15 z 6 a -2 + 9 z 6 a -4 -3 z 6 a -6 + z 6 - a 3 z 5 + 8 z 5 a -1 + 21 z 5 a -3 + 13 z 5 a -5 - z 5 a -7 + 4 a 2 z 4 -20 z 4 a -2 -9 z 4 a -4 + 6 z 4 a -6 - z 4 + 3 a 3 z 3 + 7 a z 3 -10 z 3 a -1 -27 z 3 a -3 -11 z 3 a -5 + 2 z 3 a -7 - a 2 z 2 + 11 z 2 a -2 + 4 z 2 a -4 - z 2 a -6 + 5 z 2 -3 a 3 z -5 a z + 6 z a -1 + 13 z a -3 + 5 z a -5 - a 2 -3 a -2 - a -4 -2 + a 3 z -1 + a z -1 - a -1 z -1 -2 a -3 z -1 - a -5 z -1$ (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 L10a10. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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