L10a6

(Knotscape image)

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

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_12,6,13,5 X₈₄₉₃ X_16,13,17,14 X_14,9,15,10 X_10,15,11,16 X_20,12,5,11 X_2,18,3,17
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -5, 7, -8, 9, -4, 6, -7, 8, -6, 10, -2, 3, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 4 v u 4 -4 u 4 -8 v u 3 + 8 u 3 + 8 v u 2 -8 u 2 -4 v u + 4 u + v -1$ (db)
Jones polynomial	$-q^{7/2}+4 q^{5/2}-9 q^{3/2}+13 \sqrt{q}-\frac{17}{\sqrt{q}}+\frac{17}{q^{3/2}}-\frac{17}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 -2 a 3 z 5 + 4 a z 5 - z 5 a -1 + a 5 z 3 -5 a 3 z 3 + 7 a z 3 -2 z 3 a -1 + a 5 z -4 a 3 z + 5 a z -2 z a -1 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 a 3 z 9 -2 a z 9 -6 a 4 z 8 -13 a 2 z 8 -7 z 8 -7 a 5 z 7 -16 a 3 z 7 -17 a z 7 -8 z 7 a -1 -4 a 6 z 6 + a 4 z 6 + 14 a 2 z 6 -4 z 6 a -2 + 5 z 6 - a 7 z 5 + 11 a 5 z 5 + 39 a 3 z 5 + 42 a z 5 + 14 z 5 a -1 - z 5 a -3 + 6 a 6 z 4 + 12 a 4 z 4 + 7 a 2 z 4 + 5 z 4 a -2 + 6 z 4 + a 7 z 3 -7 a 5 z 3 -28 a 3 z 3 -31 a z 3 -10 z 3 a -1 + z 3 a -3 -3 a 6 z 2 -8 a 4 z 2 -7 a 2 z 2 -2 z 2 a -2 -4 z 2 + 2 a 5 z + 8 a 3 z + 10 a z + 4 z a -1 + 1- a z -1 - a -1 z -1$ (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 L10a6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$