L10a52

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{t(1)^2 t(2)^4-2 t(1) t(2)^4+t(2)^4-3 t(1)^2 t(2)^3+6 t(1) t(2)^3-3 t(2)^3+4 t(1)^2 t(2)^2-7 t(1) t(2)^2+4 t(2)^2-3 t(1)^2 t(2)+6 t(1) t(2)-3 t(2)+t(1)^2-2 t(1)+1}{t(1) t(2)^2}$ (db)
Jones polynomial	$-4 q^{9/2}+\frac{1}{q^{9/2}}+8 q^{7/2}-\frac{4}{q^{7/2}}-12 q^{5/2}+\frac{7}{q^{5/2}}+15 q^{3/2}-\frac{12}{q^{3/2}}+q^{11/2}-16 \sqrt{q}+\frac{14}{\sqrt{q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -4 z 5 a -1 + z 5 a -3 - a 3 z 3 + 5 a z 3 -6 z 3 a -1 + 2 z 3 a -3 - a 3 z + 2 a z -3 z a -1 + z a -3 + a 3 z -1 - a z -1$ (db)
Kauffman polynomial	$-2 a z 9 -2 z 9 a -1 -5 a 2 z 8 -7 z 8 a -2 -12 z 8 -4 a 3 z 7 -9 a z 7 -15 z 7 a -1 -10 z 7 a -3 - a 4 z 6 + 10 a 2 z 6 + 2 z 6 a -2 -8 z 6 a -4 + 21 z 6 + 11 a 3 z 5 + 34 a z 5 + 40 z 5 a -1 + 13 z 5 a -3 -4 z 5 a -5 + 2 a 4 z 4 -2 a 2 z 4 + 11 z 4 a -2 + 8 z 4 a -4 - z 4 a -6 -2 z 4 -9 a 3 z 3 -27 a z 3 -27 z 3 a -1 -7 z 3 a -3 + 2 z 3 a -5 - a 4 z 2 -2 a 2 z 2 -6 z 2 a -2 -3 z 2 a -4 -4 z 2 + a 3 z + 5 a z + 6 z a -1 + 2 z a -3 - a 2 + a 3 z -1 + a z -1$ (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 L10a52. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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