L10n33

(Knotscape image)

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

Visit L10n33's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_7,16,8,17 X_20,18,5,17 X_18,11,19,12 X_10,19,11,20 X_9,14,10,15 X_15,8,16,9 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, -3, 8, -7, -6, 5, -2, 10, 7, -8, 3, 4, -5, 6, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 5 v u 2 -5 u 2 -5 v u + 5 u + v -1$ (db)
Jones polynomial	$q^{5/2}-4 q^{3/2}+6 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{8}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{2}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-2 z a 5 - a 5 z -1 + 3 z 3 a 3 + 5 z a 3 + 3 a 3 z -1 - z 5 a -2 z 3 a -3 z a -2 a z -1 + z 3 a -1$ (db)
Kauffman polynomial	$- a 4 z 8 - a 2 z 8 - a 5 z 7 -5 a 3 z 7 -4 a z 7 - a 4 z 6 -7 a 2 z 6 -6 z 6 - a 5 z 5 + 4 a 3 z 5 + a z 5 -4 z 5 a -1 -3 a 6 z 4 - a 4 z 4 + 11 a 2 z 4 - z 4 a -2 + 8 z 4 + 4 a 5 z 3 + 6 a 3 z 3 + 6 a z 3 + 4 z 3 a -1 + 5 a 6 z 2 + 6 a 4 z 2 - z 2 -4 a 5 z -9 a 3 z -5 a z - a 6 -3 a 4 -3 a 2 + a 5 z -1 + 3 a 3 z -1 + 2 a z -1$ (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 L10n33. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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