L6n1

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L6a5.gif

L6a5

L7a1.gif

L7a1

L6n1.gif Visit L6n1's page at Knotilus!

Visit L6n1's page at the original Knot Atlas!

L6n1 is [math]\displaystyle{ 6^3_3 }[/math] 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.

Knot 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 [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ \frac{w-u v}{\sqrt{u} \sqrt{v} \sqrt{w}} }[/math] (db)
Jones polynomial [math]\displaystyle{ q^4+q^2+2 }[/math] (db)
Signature 0 (db)
HOMFLY-PT polynomial [math]\displaystyle{ -z^2 a^{-2} -3 a^{-2} + a^{-4} +2-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ 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} }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{11}{6} }[/math])
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
Data:L6n1/V 2,1 Data:L6n1/V 3,1 Data:L6n1/V 4,1 Data:L6n1/V 4,2 Data:L6n1/V 4,3 Data:L6n1/V 5,1 Data:L6n1/V 5,2 Data:L6n1/V 5,3 Data:L6n1/V 5,4 Data:L6n1/V 6,1 Data:L6n1/V 6,2 Data:L6n1/V 6,3 Data:L6n1/V 6,4 Data:L6n1/V 6,5 Data:L6n1/V 6,6 Data:L6n1/V 6,7 Data:L6n1/V 6,8 Data:L6n1/V 6,9

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]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

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=3 }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

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

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math] 

In[1]:=    
 
<< KnotTheory`
 
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=
Crossings[Link[6, NonAlternating, 1]]
Out[2]=  
6
In[3]:=
PD[Link[6, NonAlternating, 1]]
Out[3]=  
PD[X[6, 1, 7, 2], X[12, 8, 9, 7], X[4, 12, 1, 11], X[5, 11, 6, 10], 
  X[3, 8, 4, 5], X[9, 3, 10, 2]]
In[4]:=
GaussCode[Link[6, NonAlternating, 1]]
Out[4]=  
GaussCode[{1, 6, -5, -3}, {-4, -1, 2, 5}, {-6, 4, 3, -2}]
In[5]:=
BR[Link[6, NonAlternating, 1]]
Out[5]=  
BR[Link[6, NonAlternating, 1]]
In[6]:=
alex = Alexander[Link[6, NonAlternating, 1]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[6, NonAlternating, 1]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[6, NonAlternating, 1]], KnotSignature[Link[6, NonAlternating, 1]]}
Out[9]=  
{Infinity, 0}
In[10]:=
J=Jones[Link[6, NonAlternating, 1]][q]
Out[10]=  
     2    4
2 + q  + q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[6, NonAlternating, 1]][q]
Out[12]=  
    2       2      4      6      8      10      12    14

3 + -- + 4 q  + 4 q  + 4 q  + 4 q  + 3 q   + 2 q   + q



    2


    q
In[13]:=
Kauffman[Link[6, NonAlternating, 1]][a, z]
Out[13]=  
                                                                2

   3    5     -2     1       2      2      2    3 z   3 z   4 z



3 + -- + -- - z   - ----- - ----- + ---- + --- - --- - --- - ---- - 



    4    2          4  2    2  2    3     a z    3     a      4
   a    a          a  z    a  z    a  z         a            a

    2    3    3    4    4
 4 z    z    z    z    z
 ---- + -- + -- + -- + --
   2     3   a     4    2


   a     a         a    a
In[14]:=
{Vassiliev[2][Link[6, NonAlternating, 1]], Vassiliev[3][Link[6, NonAlternating, 1]]}
Out[14]=  
      11

{0, -(--)}


      6
In[15]:=
Kh[Link[6, NonAlternating, 1]][q, t]
Out[15]=  
2          3          5  2    7  4    9  4

- + 3 q + q  + q t + q  t  + q  t  + q  t


q
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