L6n1

(Knotscape image)

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

Visit L6n1's page at the original Knot Atlas.

L6n1 is $6^3_3$ in Rolfsen's table of links. It makes three fibers in the Hopf fibration.

One modern form of the Germanic Valknut		Heraldic badge of Admiral Lord Boyce as Lord Warden of the Cinque Ports		Coat of arms of Suchy, Vaud, Switzerland		Rich Schwartz' "98"
Episcopal coat of arms of Dom Jacinto Bergmann (Brazil)

Planar diagram presentation	X₆₁₇₂ X_12,8,9,7 X_4,12,1,11 X_5,11,6,10 X₃₈₄₅ X_9,3,10,2
Gauss code	{1, 6, -5, -3}, {-4, -1, 2, 5}, {-6, 4, 3, -2}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{w-u v}{\sqrt{u} \sqrt{v} \sqrt{w}}$ (db)
Jones polynomial	$q 4 + q 2 + 2$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 2 a -2 -3 a -2 + a -4 + 2-2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$z 4 a -2 + z 4 a -4 + z 3 a -1 + z 3 a -3 -4 z 2 a -2 -4 z 2 a -4 -3 z a -1 -3 z a -3 + 5 a -2 + 3 a -4 + 3 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 =

0 is the signature of L6n1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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