L10n99

(Knotscape image)

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

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Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,15,8,14 X_13,5,14,8 X_11,18,12,19 X_15,20,16,17 X_19,16,20,9 X_17,12,18,13 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -9, -2, 10}, {9, -1, -3, 4}, {-8, 5, -7, 6}, {-10, 2, -5, 8, -4, 3, -6, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 3 - v u 2 + 2 v w u 2 - w u 2 + v x u 2 - v w x u 2 + w x u 2 -2 x u 2 + 2 u 2 + v u -2 v w u + w u - v x u + 2 v w x u - w x u + 2 x u - u - v w x$ (db)
Jones polynomial	$-q^{7/2}+2 q^{5/2}-6 q^{3/2}+7 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{3}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$2 a 5 z -1 + a 5 z -3 + z 3 a 3 -3 z a 3 -6 a 3 z -1 -3 a 3 z -3 - z 5 a - z 3 a + 2 z a + 5 a z -1 + 3 a z -3 + 2 z 3 a -1 + 2 z a -1 - a -1 z -3 - z a -3 - a -3 z -1$ (db)
Kauffman polynomial	$- a 2 z 8 - z 8 -4 a 3 z 7 -7 a z 7 -3 z 7 a -1 -3 a 4 z 6 -8 a 2 z 6 -2 z 6 a -2 -7 z 6 + 9 a 3 z 5 + 15 a z 5 + 5 z 5 a -1 - z 5 a -3 + 6 a 4 z 4 + 24 a 2 z 4 + 3 z 4 a -2 + 21 z 4 -6 a 5 z 3 -19 a 3 z 3 -19 a z 3 -3 z 3 a -1 + 3 z 3 a -3 -15 a 4 z 2 -33 a 2 z 2 -18 z 2 + 10 a 5 z + 23 a 3 z + 20 a z + 4 z a -1 -3 z a -3 + 13 a 4 + 24 a 2 - a -2 + 11-6 a 5 z -1 -14 a 3 z -1 -12 a z -1 -3 a -1 z -1 + a -3 z -1 -3 a 4 z -2 -6 a 2 z -2 -3 z -2 + a 5 z -3 + 3 a 3 z -3 + 3 a z -3 + a -1 z -3$ (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 L10n99. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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