L5a1

(Knotscape image)

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

Visit L5a1's page at the original Knot Atlas.

L5a1 is $5^2_1$ in Rolfsen's Table of Links. It is also known as the "Whitehead Link".

Basic depiction		Drawing of "Thor's hammer" or Mjölnir found in Sweden		Wolfgang Staubach's medallion based on this [1]
A kolam with two cycles, one of which is twisted[2]		A simplest closed-loop version of heraldic "fret" / "fretty" ornamentation.		Bisexuality symbol.

Planar diagram presentation	X₆₁₇₂ X_10,7,5,8 X₄₅₁₆ X_2,10,3,9 X₈₄₉₃
Gauss code	{1, -4, 5, -3}, {3, -1, 2, -5, 4, -2}

A Braid Representative

A Morse Link Presentation

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

-1 is the signature of L5a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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