L8n1

(Knotscape image)

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

Visit L8n1's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_9,12,10,13 X₃₈₄₉ X_5,11,6,10 X_11,5,12,16 X_13,2,14,3
Gauss code	{1, 8, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 4, -8, -2, 3, 7}

A Braid Representative

A Morse Link Presentation

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

-3 is the signature of L8n1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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