L10a21

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_18,15,19,16 X_16,10,17,9 X_8,18,9,17 X_4,19,1,20 X_12,6,13,5 X_10,4,11,3 X_20,12,5,11 X_2,14,3,13
Gauss code	{1, -10, 8, -6}, {7, -1, 2, -5, 4, -8, 9, -7, 10, -2, 3, -4, 5, -3, 6, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 4 v u 4 -4 u 4 -7 v u 3 + 7 u 3 + 7 v u 2 -7 u 2 -4 v u + 4 u + v -1$ (db)
Jones polynomial	$-q^{13/2}+4 q^{11/2}-8 q^{9/2}+12 q^{7/2}-15 q^{5/2}+16 q^{3/2}-16 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + a z 5 -4 z 5 a -1 + 2 z 5 a -3 + 2 a z 3 -6 z 3 a -1 + 5 z 3 a -3 - z 3 a -5 + a z -3 z a -1 + 3 z a -3 - z a -5 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 z 9 a -1 -2 z 9 a -3 -12 z 8 a -2 -6 z 8 a -4 -6 z 8 -7 a z 7 -12 z 7 a -1 -12 z 7 a -3 -7 z 7 a -5 -4 a 2 z 6 + 19 z 6 a -2 + 5 z 6 a -4 -4 z 6 a -6 + 6 z 6 - a 3 z 5 + 13 a z 5 + 32 z 5 a -1 + 31 z 5 a -3 + 12 z 5 a -5 - z 5 a -7 + 6 a 2 z 4 -7 z 4 a -2 + 4 z 4 a -4 + 6 z 4 a -6 + z 4 + a 3 z 3 -8 a z 3 -25 z 3 a -1 -23 z 3 a -3 -6 z 3 a -5 + z 3 a -7 - a 2 z 2 - z 2 a -2 -3 z 2 a -4 -2 z 2 a -6 - z 2 + 2 a z + 6 z a -1 + 6 z a -3 + 2 z a -5 + 1- a z -1 - a -1 z -1$ (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 L10a21. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -4$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{8}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$