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)

L5a1

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L4a1.gif

L4a1

L6a1.gif

L6a1

Contents

L5a1.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L5a1's page at Knotilus.

Visit L5a1's page at the original Knot Atlas.

L5a1 is 5^2_1 in Rolfsen's Table of Links. It is also known as the "Whitehead Link".

Basic depiction
Drawing of "Thor's hammer" or Mjölnir found in Sweden
Wolfgang Staubach's medallion based on this [1]
A kolam with two cycles, one of which is twisted[2]
A simplest closed-loop version of heraldic "fret" / "fretty" ornamentation.
Bisexuality symbol.

Link Presentations

[edit Notes on L5a1's Link Presentations]

Planar diagram presentation X6172 X10,7,5,8 X4516 X2,10,3,9 X8493
Gauss code {1, -4, 5, -3}, {3, -1, 2, -5, 4, -2}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation L5a1 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1)}{\sqrt{u} \sqrt{v}} (db)
Jones polynomial \frac{1}{q^{7/2}}-\frac{2}{q^{5/2}}-q^{3/2}+\frac{1}{q^{3/2}}+\sqrt{q}-\frac{2}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z a^3+z^3 a+2 z a+a z^{-1} -z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -z^2 a^4-2 z^3 a^3+2 z a^3-z^4 a^2-3 z^3 a+4 z a-a z^{-1} -z^4+z^2+1-z^3 a^{-1} +2 z a^{-1} - a^{-1} 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 L5a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L5a1/KhovanovTable
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2} {\mathbb Z}^{2}
r=1 {\mathbb Z}
r=2 {\mathbb Z}_2 {\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.

L4a1.gif

L4a1

L6a1.gif

L6a1