L2a1

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L2a1

L4a1

Contents

Image: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.

[edit L2a1 Quick Notes]

L2a1 is 2^2_1 in Rolfsen's table of links. It is also known as the "Hopf Link".

[edit L2a1 Further Notes and Views]


expanded Kolam Two-hearts [1]
expanded Kolam Two-hearts [1]
Are they forever linked? [2]
Are they forever linked? [2]
Three wreaths linked as two L2a1 configurations on Michelangelo's tomb
Three wreaths linked as two L2a1 configurations on Michelangelo's tomb
In Star of David form on old Jewish building in Prague
In Star of David form on old Jewish building in Prague

[edit] Link Presentations

[edit Notes on L2a1's Link Presentations]

Planar diagram presentation X4132 X2314
Gauss code {1, -2}, {2, -1}
A Braid Representative
Image:BraidPart3.gifImage:BraidPart3.gif
Image:BraidPart4.gifImage:BraidPart4.gif
A Morse Link Presentation Image:L2a1_ML.gif

[edit] 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 a3z−1−za−az−1 (db)
Kauffman polynomial −za3 + a3z−1−a2−za + az−1 (db)

[edit] Khovanov Homology

The coefficients of the monomials trqj are shown, along with their alternating sums χ (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.    Data:L2a1/KhovanovTable
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}

[edit] Computer Talk

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

[edit] Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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L2a1

L4a1

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