L10n16

(Knotscape image)

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

Visit L10n16's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_9,14,10,15 X₈₄₉₃ X_12,5,13,6 X_20,13,5,14 X_11,16,12,17 X_15,10,16,11 X_2,18,3,17
Gauss code	{1, -10, 5, -3}, {6, -1, 2, -5, -4, 9, -8, -6, 7, 4, -9, 8, 10, -2, 3, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 u 3 + 5 v u 2 -5 u 2 -5 v u + 5 u + 2 v -2$ (db)
Jones polynomial	$-\frac{3}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z a 9 -3 z 3 a 7 -5 z a 7 -2 a 7 z -1 + 2 z 5 a 5 + 7 z 3 a 5 + 10 z a 5 + 5 a 5 z -1 -3 z 3 a 3 -6 z a 3 -3 a 3 z -1$ (db)
Kauffman polynomial	$- z 5 a 11 + z 3 a 11 -4 z 6 a 10 + 8 z 4 a 10 -3 z 2 a 10 - a 10 -5 z 7 a 9 + 9 z 5 a 9 -3 z 3 a 9 + 2 z a 9 -2 z 8 a 8 -6 z 6 a 8 + 20 z 4 a 8 -10 z 2 a 8 -10 z 7 a 7 + 21 z 5 a 7 -17 z 3 a 7 + 9 z a 7 -2 a 7 z -1 -2 z 8 a 6 -5 z 6 a 6 + 15 z 4 a 6 -14 z 2 a 6 + 5 a 6 -5 z 7 a 5 + 11 z 5 a 5 -19 z 3 a 5 + 15 z a 5 -5 a 5 z -1 -3 z 6 a 4 + 3 z 4 a 4 -7 z 2 a 4 + 5 a 4 -6 z 3 a 3 + 8 z a 3 -3 a 3 z -1$ (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 =

-3 is the signature of L10n16. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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