L10a79

(Knotscape image)

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Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_20,13,7,14 X_14,9,15,10 X_10,19,11,20 X_16,6,17,5 X_18,16,19,15 X₂₇₃₈ X_4,11,5,12 X_6,18,1,17
Gauss code	{1, -8, 2, -9, 6, -10}, {8, -1, 4, -5, 9, -2, 3, -4, 7, -6, 10, -7, 5, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 + u 4 -2 v 2 u 3 + 5 v u 3 -4 u 3 + 5 v 2 u 2 -9 v u 2 + 5 u 2 -4 v 2 u + 5 v u -2 u + v 2 - v$ (db)
Jones polynomial	$q^{5/2}-4 q^{3/2}+8 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{14}{q^{3/2}}-\frac{16}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + 3 z 3 a 5 + 5 z a 5 + 3 a 5 z -1 -2 z 5 a 3 -5 z 3 a 3 -6 z a 3 -2 a 3 z -1 - z 5 a + z a + z 3 a -1$ (db)
Kauffman polynomial	$-2 a 5 z 9 -2 a 3 z 9 -4 a 6 z 8 -11 a 4 z 8 -7 a 2 z 8 -3 a 7 z 7 -6 a 5 z 7 -13 a 3 z 7 -10 a z 7 - a 8 z 6 + 7 a 6 z 6 + 20 a 4 z 6 + 4 a 2 z 6 -8 z 6 + 8 a 7 z 5 + 26 a 5 z 5 + 36 a 3 z 5 + 14 a z 5 -4 z 5 a -1 + 3 a 8 z 4 + 2 a 6 z 4 -3 a 4 z 4 + 7 a 2 z 4 - z 4 a -2 + 8 z 4 -7 a 7 z 3 -24 a 5 z 3 -27 a 3 z 3 -8 a z 3 + 2 z 3 a -1 -3 a 8 z 2 -7 a 6 z 2 -6 a 4 z 2 -4 a 2 z 2 -2 z 2 + 3 a 7 z + 10 a 5 z + 9 a 3 z + 2 a z + a 8 + 3 a 6 + 3 a 4 - a 7 z -1 -3 a 5 z -1 -2 a 3 z -1$ (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 L10a79. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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