L10a78

(Knotscape image)

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,8,15,7 X_6,13,1,14 X_18,12,19,11 X_16,5,17,6 X_12,18,13,17 X_20,16,7,15 X_4,19,5,20
Gauss code	{1, -2, 3, -10, 7, -5}, {4, -1, 2, -3, 6, -8, 5, -4, 9, -7, 8, -6, 10, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 4 v 2 u 3 -4 v u 3 -4 v 2 u 2 + 7 v u 2 -4 u 2 -4 v u + 4 u -1$ (db)
Jones polynomial	$-q^{5/2}+3 q^{3/2}-5 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{1}{q^{15/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 3 z 7 + a 5 z 5 -5 a 3 z 5 + 2 a z 5 + 3 a 5 z 3 -10 a 3 z 3 + 7 a z 3 - z 3 a -1 + 3 a 5 z -10 a 3 z + 6 a z -2 z a -1 + 2 a 5 z -1 -3 a 3 z -1 + a z -1$ (db)
Kauffman polynomial	$- z 3 a 9 -3 z 4 a 8 -6 z 5 a 7 + 5 z 3 a 7 -2 z a 7 -8 z 6 a 6 + 10 z 4 a 6 -2 z 2 a 6 -9 z 7 a 5 + 19 z 5 a 5 -13 z 3 a 5 + 7 z a 5 -2 a 5 z -1 -6 z 8 a 4 + 10 z 6 a 4 + 4 z 4 a 4 -7 z 2 a 4 + 3 a 4 -2 z 9 a 3 -5 z 7 a 3 + 31 z 5 a 3 -31 z 3 a 3 + 12 z a 3 -3 a 3 z -1 -9 z 8 a 2 + 31 z 6 a 2 -26 z 4 a 2 + z 2 a 2 + 3 a 2 -2 z 9 a + 3 z 7 a + 10 z 5 a -17 z 3 a + 5 z a - a z -1 -3 z 8 + 13 z 6 -17 z 4 + 6 z 2 + 1- z 7 a -1 + 4 z 5 a -1 -5 z 3 a -1 + 2 z a -1$ (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 L10a78. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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