# L5a1

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L5a1 at Knotilus! L5a1 is $5^2_1$ in Rolfsen's Table of Links. It is also known as the "Whitehead Link".  Drawing of "Thor's hammer" or Mjölnir found in Sweden  Wolfgang Staubach's medallion based on this  A kolam with two cycles, one of which is twisted  A simplest closed-loop version of heraldic "fret" / "fretty" ornamentation.

### 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 $r$).
 \ r \ j \
-3-2-1012χ
4     11
2      0
0   21 1
-2  12  1
-4 1    1
-6 1    1
-81     -1
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.