L2a1

L2a1

L4a1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L2a1 at Knotilus!
	L2a1 is [math]\displaystyle{ 2^2_1 }[/math] 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

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 [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -1 }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{\sqrt{q}}-\frac{1}{q^{5/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^{-1} -z a-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z a^3+a^3 z^{-1} -a^2-z a+a z^{-1} }[/math] (db)

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]).

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

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

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