# L6n1

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L6n1 at Knotilus! L6n1 is ${\displaystyle 6_{3}^{3}}$ in Rolfsen's table of links. It makes three fibers in the Hopf fibration.
 One modern form of the Germanic Valknut Basic depiction Basic symmetrical depiction Three squares, as impossible object Coat of arms of Suchy, Vaud, Switzerland Rich Schwartz' "98" 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.

### Link Presentations

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

### Computer Talk

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

### Modifying This Page

Read me first: Modifying Knot Pages

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

See/edit the Link_Splice_Base (expert).

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