L10a20

(Knotscape image)

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

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	$q^{11/2}-4 q^{9/2}+9 q^{7/2}-13 q^{5/2}+15 q^{3/2}-17 \sqrt{q}+\frac{14}{\sqrt{q}}-\frac{12}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -4 z 5 a -1 + z 5 a -3 - a 3 z 3 + 6 a z 3 -7 z 3 a -1 + 2 z 3 a -3 -2 a 3 z + 7 a z -7 z a -1 + 2 z a -3 - a 3 z -1 + 4 a z -1 -4 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$-2 a z 9 -2 z 9 a -1 -4 a 2 z 8 -8 z 8 a -2 -12 z 8 -3 a 3 z 7 -8 a z 7 -17 z 7 a -1 -12 z 7 a -3 - a 4 z 6 + 6 a 2 z 6 + 3 z 6 a -2 -9 z 6 a -4 + 19 z 6 + 8 a 3 z 5 + 31 a z 5 + 45 z 5 a -1 + 18 z 5 a -3 -4 z 5 a -5 + 3 a 4 z 4 + 4 a 2 z 4 + 14 z 4 a -2 + 9 z 4 a -4 - z 4 a -6 + 5 z 4 -8 a 3 z 3 -29 a z 3 -33 z 3 a -1 -11 z 3 a -3 + z 3 a -5 -3 a 4 z 2 -10 a 2 z 2 -12 z 2 a -2 -4 z 2 a -4 -15 z 2 + 4 a 3 z + 13 a z + 13 z a -1 + 4 z a -3 + a 4 + 4 a 2 + 4 a -2 + a -4 + 7- a 3 z -1 -4 a z -1 -4 a -1 z -1 - a -3 z -1$ (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 L10a20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$