L10a53

(Knotscape image)

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Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X₆₇₁₈ X_20,14,7,13 X_12,5,13,6 X_10,4,11,3 X_4,15,5,16 X_16,12,17,11 X_14,20,15,19 X_2,18,3,17
Gauss code	{1, -10, 6, -7, 5, -3}, {3, -1, 2, -6, 8, -5, 4, -9, 7, -8, 10, -2, 9, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + 4 v 2 u 3 -6 v u 3 + 2 u 3 -5 v 2 u 2 + 11 v u 2 -5 u 2 + 2 v 2 u -6 v u + 4 u + v -1$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+8 q^{5/2}-12 q^{3/2}+15 \sqrt{q}-\frac{17}{\sqrt{q}}+\frac{15}{q^{3/2}}-\frac{13}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 4 a z 5 -2 z 5 a -1 -2 a 3 z 3 + 6 a z 3 -5 z 3 a -1 + z 3 a -3 - a 3 z + 2 a z -3 z a -1 + z a -3 + a 3 z -1 - a z -1$ (db)
Kauffman polynomial	$-3 a z 9 -3 z 9 a -1 -9 a 2 z 8 -6 z 8 a -2 -15 z 8 -11 a 3 z 7 -12 a z 7 -5 z 7 a -1 -4 z 7 a -3 -8 a 4 z 6 + 11 a 2 z 6 + 14 z 6 a -2 - z 6 a -4 + 34 z 6 -4 a 5 z 5 + 16 a 3 z 5 + 39 a z 5 + 29 z 5 a -1 + 10 z 5 a -3 - a 6 z 4 + 7 a 4 z 4 -2 a 2 z 4 -7 z 4 a -2 + 2 z 4 a -4 -19 z 4 + 2 a 5 z 3 -9 a 3 z 3 -29 a z 3 -25 z 3 a -1 -7 z 3 a -3 - a 4 z 2 - z 2 a -4 + 2 z 2 + a 3 z + 5 a z + 6 z a -1 + 2 z a -3 - a 2 + a 3 z -1 + a z -1$ (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 L10a53. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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