L10a2

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_14,10,15,9 X₈₄₉₃ X_10,5,11,6 X_20,13,5,14 X_18,11,19,12 X_12,19,13,20 X_2,16,3,15
Gauss code	{1, -10, 5, -3}, {6, -1, 2, -5, 4, -6, 8, -9, 7, -4, 10, -2, 3, -8, 9, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 4 v u 4 -4 u 4 -6 v u 3 + 6 u 3 + 6 v u 2 -6 u 2 -4 v u + 4 u + v -1$ (db)
Jones polynomial	$q^{3/2}-4 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{13}{q^{5/2}}-\frac{15}{q^{7/2}}+\frac{14}{q^{9/2}}-\frac{11}{q^{11/2}}+\frac{8}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 7 - z a 7 + 2 z 5 a 5 + 5 z 3 a 5 + 2 z a 5 - a 5 z -1 - z 7 a 3 -4 z 5 a 3 -5 z 3 a 3 + 3 a 3 z -1 + z 5 a + 2 z 3 a - z a -2 a z -1$ (db)
Kauffman polynomial	$- z 4 a 10 -4 z 5 a 9 + 2 z 3 a 9 -8 z 6 a 8 + 9 z 4 a 8 -3 z 2 a 8 -9 z 7 a 7 + 11 z 5 a 7 -5 z 3 a 7 + 2 z a 7 -6 z 8 a 6 + z 6 a 6 + 9 z 4 a 6 -2 z 2 a 6 - a 6 -2 z 9 a 5 -11 z 7 a 5 + 30 z 5 a 5 -20 z 3 a 5 + 4 z a 5 + a 5 z -1 -11 z 8 a 4 + 23 z 6 a 4 -12 z 4 a 4 + 4 z 2 a 4 -3 a 4 -2 z 9 a 3 -6 z 7 a 3 + 27 z 5 a 3 -22 z 3 a 3 + z a 3 + 3 a 3 z -1 -5 z 8 a 2 + 13 z 6 a 2 -9 z 4 a 2 + 3 z 2 a 2 -3 a 2 -4 z 7 a + 12 z 5 a -9 z 3 a - z a + 2 a z -1 - z 6 + 2 z 4$ (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 L10a2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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