L10a84

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

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Planar diagram presentation	X₈₁₉₂ X_14,9,15,10 X₄₇₅₈ X_16,6,17,5 X_18,16,19,15 X_6,18,1,17 X_20,11,7,12 X_10,19,11,20 X_2,14,3,13 X_12,4,13,3
Gauss code	{1, -9, 10, -3, 4, -6}, {3, -1, 2, -8, 7, -10, 9, -2, 5, -4, 6, -5, 8, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 2 v u 4 - u 4 + 2 v 2 u 3 -6 v u 3 + 3 u 3 -3 v 2 u 2 + 7 v u 2 -3 u 2 + 3 v 2 u -6 v u + 2 u - v 2 + 2 v -1$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+7 q^{5/2}-11 q^{3/2}+13 \sqrt{q}-\frac{15}{\sqrt{q}}+\frac{13}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 5 a z 5 -2 z 5 a -1 -3 a 3 z 3 + 11 a z 3 -6 z 3 a -1 + z 3 a -3 -4 a 3 z + 10 a z -6 z a -1 + z a -3 - a 3 z -1 + 3 a z -1 -2 a -1 z -1$ (db)
Kauffman polynomial	$-2 a z 9 -2 z 9 a -1 -6 a 2 z 8 -5 z 8 a -2 -11 z 8 -8 a 3 z 7 -11 a z 7 -7 z 7 a -1 -4 z 7 a -3 -6 a 4 z 6 + 4 a 2 z 6 + 11 z 6 a -2 - z 6 a -4 + 22 z 6 -3 a 5 z 5 + 13 a 3 z 5 + 35 a z 5 + 30 z 5 a -1 + 11 z 5 a -3 - a 6 z 4 + 7 a 4 z 4 + 7 a 2 z 4 -3 z 4 a -2 + 2 z 4 a -4 -6 z 4 + 2 a 5 z 3 -13 a 3 z 3 -33 a z 3 -26 z 3 a -1 -8 z 3 a -3 + a 6 z 2 -5 a 4 z 2 -10 a 2 z 2 -2 z 2 a -2 - z 2 a -4 -5 z 2 + 6 a 3 z + 15 a z + 10 z a -1 + z a -3 + a 4 + 3 a 2 + 3- a 3 z -1 -3 a z -1 -2 a -1 z -1$ (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 L10a84. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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