L10a70

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 2 v u 4 - u 4 + 3 v 2 u 3 -7 v u 3 + 4 u 3 -5 v 2 u 2 + 9 v u 2 -5 u 2 + 4 v 2 u -7 v u + 3 u - v 2 + 2 v -1$ (db)
Jones polynomial	$q^{9/2}-5 q^{7/2}+9 q^{5/2}-14 q^{3/2}+17 \sqrt{q}-\frac{19}{\sqrt{q}}+\frac{17}{q^{3/2}}-\frac{14}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 4 a z 5 -2 z 5 a -1 -2 a 3 z 3 + 7 a z 3 -4 z 3 a -1 + z 3 a -3 -2 a 3 z + 4 a z - z a -1 + a -1 z -1 - a -3 z -1$ (db)
Kauffman polynomial	$-4 a z 9 -4 z 9 a -1 -11 a 2 z 8 -8 z 8 a -2 -19 z 8 -13 a 3 z 7 -13 a z 7 -5 z 7 a -1 -5 z 7 a -3 -9 a 4 z 6 + 13 a 2 z 6 + 19 z 6 a -2 - z 6 a -4 + 42 z 6 -4 a 5 z 5 + 19 a 3 z 5 + 42 a z 5 + 30 z 5 a -1 + 11 z 5 a -3 - a 6 z 4 + 8 a 4 z 4 - a 2 z 4 -11 z 4 a -2 + z 4 a -4 -22 z 4 + a 5 z 3 -12 a 3 z 3 -27 a z 3 -19 z 3 a -1 -5 z 3 a -3 -3 a 4 z 2 - a 2 z 2 + z 2 a -2 + 3 z 2 + 3 a 3 z + 5 a z -2 z a -3 - a -2 + a -1 z -1 + a -3 z -1$ (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 L10a70. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 2$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = 3$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$