L5a1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L5a1 at Knotilus!
	[edit L5a1 Quick Notes] L5a1 is $5_{1}^{2}$ 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	$-za^{3}+z^{3}a+2za+az^{-1}-za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{2}a^{4}-2z^{3}a^{3}+2za^{3}-z^{4}a^{2}-3z^{3}a+4za-az^{-1}-z^{4}+z^{2}+1-z^{3}a^{-1}+2za^{-1}-a^{-1}z^{-1}$ (db)

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

-3

-2

-1

0

1

2

χ

4

1

2

0

2

1

-2

1

2

1

-4

1

-6

1

-8

1

-1

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