L10n98

(Knotscape image)

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

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Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_13,15,14,20 X_10,3,11,4 X_4,9,1,10 X_16,7,17,8 X_8,15,5,16 X_11,19,12,18 X_19,13,20,12 X_17,9,18,14
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, -8, 9, -3, 10}, {7, -6, -10, 8, -9, 3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + v 2 w u 2 + v 2 x u 2 - v x u 2 - v 2 w u + v w u + v x u - x u - v w + w - w x + x$ (db)
Jones polynomial	$-2 q^{5/2}+2 q^{3/2}-4 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z a 5 + 3 a 5 z -1 + a 5 z -3 -3 z 3 a 3 -11 z a 3 -10 a 3 z -1 -3 a 3 z -3 + 2 z 5 a + 10 z 3 a + 16 z a + 11 a z -1 + 3 a z -3 -2 z 3 a -1 -6 z a -1 -4 a -1 z -1 - a -1 z -3$ (db)
Kauffman polynomial	$- a 4 z 8 - a 2 z 8 - a 5 z 7 -5 a 3 z 7 -4 a z 7 + 3 a 4 z 6 - a 2 z 6 -4 z 6 + 6 a 5 z 5 + 25 a 3 z 5 + 17 a z 5 -2 z 5 a -1 + 4 a 4 z 4 + 19 a 2 z 4 - z 4 a -2 + 14 z 4 -13 a 5 z 3 -39 a 3 z 3 -24 a z 3 + 2 z 3 a -1 -17 a 4 z 2 -33 a 2 z 2 -16 z 2 + 13 a 5 z + 28 a 3 z + 21 a z + 3 z a -1 -3 z a -3 + 13 a 4 + 24 a 2 - a -2 + 11-6 a 5 z -1 -14 a 3 z -1 -12 a z -1 -3 a -1 z -1 + a -3 z -1 -3 a 4 z -2 -6 a 2 z -2 -3 z -2 + a 5 z -3 + 3 a 3 z -3 + 3 a z -3 + a -1 z -3$ (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 L10n98. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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