L8n3

(Knotscape image)

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

Visit L8n3's page at the original Knot Atlas.

L8n3 is $8^3_{7}$ in the Rolfsen table of links.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{u v^2 w^2-1}{\sqrt{u} v w}$ (db)
Jones polynomial	$q -3 + q -5 + q -7 + q -9$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a 10 z -2 + a 10 - z 4 a 8 -5 z 2 a 8 -2 a 8 z -2 -6 a 8 + z 6 a 6 + 6 z 4 a 6 + 10 z 2 a 6 + a 6 z -2 + 5 a 6$ (db)
Kauffman polynomial	$a 12 + a 10 z 2 + a 10 z -2 -3 a 10 + a 9 z 5 -5 a 9 z 3 + 6 a 9 z -2 a 9 z -1 + a 8 z 6 -6 a 8 z 4 + 11 a 8 z 2 + 2 a 8 z -2 -8 a 8 + a 7 z 5 -5 a 7 z 3 + 6 a 7 z -2 a 7 z -1 + a 6 z 6 -6 a 6 z 4 + 10 a 6 z 2 + a 6 z -2 -5 a 6$ (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 =

-6 is the signature of L8n3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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