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)

# L6n1

## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L6n1's page at Knotilus. Visit L6n1's page at the original Knot Atlas. 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 Basic depiction Coat of arms of Suchy, Vaud, Switzerland Rich Schwartz' "98" Episcopal coat of arms of Dom Jacinto Bergmann (Brazil) Basic symmetrical depiction Heraldic badge of Admiral Lord Boyce as Lord Warden of the Cinque Ports Canadian trade-union federation emblem.

 Planar diagram presentation X6172 X12,8,9,7 X4,12,1,11 X5,11,6,10 X3845 X9,3,10,2 Gauss code {1, 6, -5, -3}, {-4, -1, 2, 5}, {-6, 4, 3, -2}

### 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 $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of L6n1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ 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.