L9n13

(Knotscape image)

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

Visit L9n13's page at the original Knot Atlas.

Planar diagram presentation	X₈₁₉₂ X_11,17,12,16 X_3,10,4,11 X_15,3,16,2 X_5,13,6,12 X₆₇₁₈ X_9,14,10,15 X_13,18,14,7 X_17,4,18,5
Gauss code	{1, 4, -3, 9, -5, -6}, {6, -1, -7, 3, -2, 5, -8, 7, -4, 2, -9, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + v u 2 + v 2 u -3 v u + u + v -1$ (db)
Jones polynomial	$\sqrt{q}-\frac{3}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 5 + 2 z a 5 - z 5 a 3 -4 z 3 a 3 -4 z a 3 + a 3 z -1 + z 3 a + z a - a z -1$ (db)
Kauffman polynomial	$- z 5 a 7 + 3 z 3 a 7 - z a 7 -2 z 6 a 6 + 7 z 4 a 6 -5 z 2 a 6 - z 7 a 5 + 2 z 5 a 5 + z a 5 -3 z 6 a 4 + 9 z 4 a 4 -7 z 2 a 4 - z 7 a 3 + 3 z 5 a 3 -6 z 3 a 3 + 4 z a 3 + a 3 z -1 - z 6 a 2 + 2 z 4 a 2 -3 z 2 a 2 - a 2 -3 z 3 a + 2 z a + a z -1 - z 2$ (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 L9n13. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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