L10n13

(Knotscape image)

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

Visit L10n13'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 X₃₈₄₉ X_5,13,6,12 X_13,5,14,20 X_11,16,12,17 X_15,10,16,11 X_17,2,18,3
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 $u$ , $v$ , $w$ , ...)	$- v u 5 + 2 v u 4 - v u 3 + u 3 + v u 2 - u 2 + 2 u -1$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z a 9 + a 9 z -1 - z 5 a 7 -5 z 3 a 7 -7 z a 7 -3 a 7 z -1 + z 7 a 5 + 6 z 5 a 5 + 12 z 3 a 5 + 11 z a 5 + 4 a 5 z -1 - z 5 a 3 -5 z 3 a 3 -7 z a 3 -2 a 3 z -1$ (db)
Kauffman polynomial	$- z a 11 -2 z 2 a 10 + a 10 - z 5 a 9 + z 3 a 9 + z a 9 - a 9 z -1 -3 z 6 a 8 + 11 z 4 a 8 -10 z 2 a 8 + 3 a 8 -3 z 7 a 7 + 13 z 5 a 7 -17 z 3 a 7 + 11 z a 7 -3 a 7 z -1 - z 8 a 6 + z 6 a 6 + 9 z 4 a 6 -12 z 2 a 6 + 3 a 6 -4 z 7 a 5 + 20 z 5 a 5 -30 z 3 a 5 + 18 z a 5 -4 a 5 z -1 - z 8 a 4 + 4 z 6 a 4 -2 z 4 a 4 -4 z 2 a 4 + 2 a 4 - z 7 a 3 + 6 z 5 a 3 -12 z 3 a 3 + 9 z a 3 -2 a 3 z -1$ (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 =

-5 is the signature of L10n13. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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