# L5a1

## Contents

 (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.

 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}

### 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 −za3 + z3a + 2za + az−1−za−1−a−1z−1 (db) Kauffman polynomial −z2a4−2z3a3 + 2za3−z4a2−3z3a + 4za−az−1−z4 + z2 + 1−z3a−1 + 2za−1−a−1z−1 (db)

### 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 L5a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. Data:L5a1/KhovanovTable
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ 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.

Read me first: Modifying Knot Pages

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