Notice. The Knot Atlas is now recovering from a major crash. Hopefully all functionality will return slowly over the next few days. --Drorbn (talk) 21:23, 4 July 2013 (EDT)

L2a1

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L2a1.gif

L2a1

L4a1.gif

L4a1

Contents

L2a1.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L2a1's page at Knotilus.

Visit L2a1's page at the original Knot Atlas.

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
BraidPart3.gifBraidPart3.gif
BraidPart4.gifBraidPart4.gif
A Morse Link Presentation L2a1 ML.gif

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). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=-1 is the signature of L2a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10χ
0  11
-2  11
-41  1
-61  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.gif

L2a1

L4a1.gif

L4a1