L10a9

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_18,14,19,13 X_16,9,17,10 X_8,17,9,18 X_20,16,5,15 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, 4, -8, 7, -5, 6, -4, 8, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 3 v u 4 -3 u 4 -4 v u 3 + 4 u 3 + 4 v u 2 -4 u 2 -3 v u + 3 u + v -1$ (db)
Jones polynomial	$-q^{15/2}+3 q^{13/2}-6 q^{11/2}+9 q^{9/2}-10 q^{7/2}+10 q^{5/2}-10 q^{3/2}+7 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 -2 z 5 a -1 + 5 z 5 a -3 - z 5 a -5 + a z 3 -8 z 3 a -1 + 10 z 3 a -3 -3 z 3 a -5 + 3 a z -10 z a -1 + 10 z a -3 -3 z a -5 + 2 a z -1 -4 a -1 z -1 + 3 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -7 z 8 a -2 -5 z 8 a -4 -2 z 8 - a z 7 -3 z 7 a -1 -11 z 7 a -3 -9 z 7 a -5 + 21 z 6 a -2 + 4 z 6 a -4 -9 z 6 a -6 + 8 z 6 + 5 a z 5 + 25 z 5 a -1 + 44 z 5 a -3 + 18 z 5 a -5 -6 z 5 a -7 -11 z 4 a -2 + 15 z 4 a -4 + 14 z 4 a -6 -3 z 4 a -8 -9 z 4 -9 a z 3 -39 z 3 a -1 -45 z 3 a -3 -10 z 3 a -5 + 4 z 3 a -7 - z 3 a -9 -6 z 2 a -2 -14 z 2 a -4 -7 z 2 a -6 + z 2 + 7 a z + 21 z a -1 + 19 z a -3 + 4 z a -5 - z a -7 + 3 a -2 + 3 a -4 + a -6 + 2-2 a z -1 -4 a -1 z -1 -3 a -3 z -1 - a -5 z -1$ (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 =

3 is the signature of L10a9. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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