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 also known as the "Hopf Link".

[edit L2a1 Further Notes and Views]
exapnded Kolam Two-hearts [1]
exapnded Kolam Two-hearts [1]
Are they forever linked? [2]
Are they forever linked? [2]

[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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