L2a1

L2a1

L4a1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L2a1 at Knotilus!
	L2a1 is $2^2_1$ in Rolfsen's table of links. It is also known as 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

Link Presentations

[edit Notes on L2a1's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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	$a^3 z^{-1} -z a-a z^{-1}$ (db)
Kauffman polynomial	$-z a^3+a^3 z^{-1} -a^2-z a+a z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -2 -1 0 χ 0     1 1 -2     1 1 -4 1     1 -6 1     1

Integral Khovanov Homology (db, data source)

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

L2a1 L4a1