# L6n1

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L6n1 at Knotilus! L6n1 is $6^3_3$ in Rolfsen's table of links. It makes three fibers in the Hopf fibration.  One modern form of the Germanic Valknut  Coat of arms of Suchy, Vaud, Switzerland  Episcopal coat of arms of Dom Jacinto Bergmann (Brazil)  Heraldic badge of Admiral Lord Boyce as Lord Warden of the Cinque Ports  Canadian trade-union federation emblem.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{w-u v}{\sqrt{u} \sqrt{v} \sqrt{w}}$ (db) Jones polynomial $q^4+q^2+2$ (db) Signature 0 (db) HOMFLY-PT polynomial $-z^2 a^{-2} -3 a^{-2} + a^{-4} +2-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2}$ (db) Kauffman polynomial $z^4 a^{-2} +z^4 a^{-4} +z^3 a^{-1} +z^3 a^{-3} -4 z^2 a^{-2} -4 z^2 a^{-4} -3 z a^{-1} -3 z a^{-3} +5 a^{-2} +3 a^{-4} +3+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-2} - z^{-2}$ (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 \
01234χ
9    11
7    11
5  1  1
31    1
131   2
-12    2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $i=3$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ $r=4$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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