L2a1

(Knotscape image)

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

L2a1 is $2_{1}^{2}$ in Rolfsen's table of links. It is also known as the "Hopf Link".

The sheet bend of practical knot tying deforms to the Hopf link.

Japanese family emblem	Linked hearts	expanded Kolam Two-hearts [1]	Logo
In Star of David form on old Jewish building in Prague	Three wreaths linked as two L2a1 configurations on Michelangelo's tomb	One form of a heterosexuality symbol	(pseudo-3D)
Are they forever linked? [2]	As impossible object	Linked hearts (pseudo-3D)	German coat of arms

Planar diagram presentation	X₄₁₃₂ X₂₃₁₄
Gauss code	{1, -2}, {2, -1}

A Braid Representative

A Morse Link Presentation

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

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

-2

-1

0

χ

0

1

-2

1

-4

1

-6

1

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