L10a13

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 5 -2 v u 4 + 4 u 4 + 5 v u 3 -7 u 3 -7 v u 2 + 5 u 2 + 4 v u -2 u - v$ (db)
Jones polynomial	$q^{5/2}-3 q^{3/2}+6 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{13}{q^{5/2}}+\frac{11}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + 3 z 3 a 5 + 6 z a 5 + 4 a 5 z -1 -2 z 5 a 3 -6 z 3 a 3 -8 z a 3 -4 a 3 z -1 - z 5 a - z 3 a + a z -1 + z 3 a -1 + z a -1$ (db)
Kauffman polynomial	$- a 5 z 9 - a 3 z 9 -3 a 6 z 8 -7 a 4 z 8 -4 a 2 z 8 -3 a 7 z 7 -9 a 5 z 7 -12 a 3 z 7 -6 a z 7 - a 8 z 6 + 3 a 6 z 6 + 7 a 4 z 6 -2 a 2 z 6 -5 z 6 + 9 a 7 z 5 + 32 a 5 z 5 + 33 a 3 z 5 + 7 a z 5 -3 z 5 a -1 + 3 a 8 z 4 + 10 a 6 z 4 + 16 a 4 z 4 + 15 a 2 z 4 - z 4 a -2 + 5 z 4 -9 a 7 z 3 -32 a 5 z 3 -31 a 3 z 3 -5 a z 3 + 3 z 3 a -1 -3 a 8 z 2 -13 a 6 z 2 -21 a 4 z 2 -15 a 2 z 2 + z 2 a -2 -3 z 2 + 4 a 7 z + 17 a 5 z + 16 a 3 z + 2 a z - z a -1 + a 8 + 4 a 6 + 7 a 4 + 4 a 2 + 1- a 7 z -1 -4 a 5 z -1 -4 a 3 z -1 - a z -1$ (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 L10a13. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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