# L5a1

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L5a1 at Knotilus! L5a1 is ${\displaystyle 5_{1}^{2}}$ 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 ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(u-1)(v-1)}{{\sqrt {u}}{\sqrt {v}}}}}$ (db) Jones polynomial ${\displaystyle {\frac {1}{q^{7/2}}}-{\frac {2}{q^{5/2}}}-q^{3/2}+{\frac {1}{q^{3/2}}}+{\sqrt {q}}-{\frac {2}{\sqrt {q}}}}$ (db) Signature -1 (db) HOMFLY-PT polynomial ${\displaystyle -za^{3}+z^{3}a+2za+az^{-1}-za^{-1}-a^{-1}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle -z^{2}a^{4}-2z^{3}a^{3}+2za^{3}-z^{4}a^{2}-3z^{3}a+4za-az^{-1}-z^{4}+z^{2}+1-z^{3}a^{-1}+2za^{-1}-a^{-1}z^{-1}}$ (db)

### Khovanov Homology

The coefficients of the monomials ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$).
 \ 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
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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