L10a81

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 -2 v 2 u 3 + 6 v u 3 -2 u 3 + 5 v 2 u 2 -9 v u 2 + 5 u 2 -2 v 2 u + 6 v u -2 u - v$ (db)
Jones polynomial	$-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{13}{q^{7/2}}+\frac{14}{q^{9/2}}-\frac{13}{q^{11/2}}+\frac{10}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z a 9 + a 9 z -1 -3 z 3 a 7 -5 z a 7 -2 a 7 z -1 + 2 z 5 a 5 + 5 z 3 a 5 + 5 z a 5 + 2 a 5 z -1 + z 5 a 3 -3 z a 3 - a 3 z -1 - z 3 a - z a$ (db)
Kauffman polynomial	$- z 5 a 11 + 2 z 3 a 11 - z a 11 -3 z 6 a 10 + 5 z 4 a 10 - z 2 a 10 -5 z 7 a 9 + 9 z 5 a 9 -6 z 3 a 9 + 5 z a 9 - a 9 z -1 -5 z 8 a 8 + 8 z 6 a 8 -5 z 4 a 8 + 2 z 2 a 8 -2 z 9 a 7 -6 z 7 a 7 + 22 z 5 a 7 -26 z 3 a 7 + 12 z a 7 -2 a 7 z -1 -10 z 8 a 6 + 22 z 6 a 6 -20 z 4 a 6 + 6 z 2 a 6 - a 6 -2 z 9 a 5 -6 z 7 a 5 + 22 z 5 a 5 -26 z 3 a 5 + 12 z a 5 -2 a 5 z -1 -5 z 8 a 4 + 8 z 6 a 4 -5 z 4 a 4 + 2 z 2 a 4 -5 z 7 a 3 + 9 z 5 a 3 -6 z 3 a 3 + 5 z a 3 - a 3 z -1 -3 z 6 a 2 + 5 z 4 a 2 - z 2 a 2 - z 5 a + 2 z 3 a - z a$ (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 =

-3 is the signature of L10a81. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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