L7a6

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L7a5.gif

L7a5

L7a7.gif

L7a7

L7a6.gif Visit L7a6's page at Knotilus!

Visit L7a6's page at the original Knot Atlas!

L7a6 is in the Rolfsen table of links.


L7a6 Further Notes and Views

Knot presentations

Planar diagram presentation X8192 X10,4,11,3 X14,10,7,9 X12,6,13,5 X2738 X4,12,5,11 X6,14,1,13
Gauss code {1, -5, 2, -6, 4, -7}, {5, -1, 3, -2, 6, -4, 7, -3}

Polynomial invariants

Multivariable Alexander Polynomial (in , , , ...) (db)
Jones polynomial (db)
Signature 3 (db)
HOMFLY-PT polynomial (db)
Kauffman polynomial (db)

Vassiliev invariants

V2 and V3: (0, )
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:L7a6/V 2,1 Data:L7a6/V 3,1 Data:L7a6/V 4,1 Data:L7a6/V 4,2 Data:L7a6/V 4,3 Data:L7a6/V 5,1 Data:L7a6/V 5,2 Data:L7a6/V 5,3 Data:L7a6/V 5,4 Data:L7a6/V 6,1 Data:L7a6/V 6,2 Data:L7a6/V 6,3 Data:L7a6/V 6,4 Data:L7a6/V 6,5 Data:L7a6/V 6,6 Data:L7a6/V 6,7 Data:L7a6/V 6,8 Data:L7a6/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 are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 3 is the signature of L7a6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-1012345χ
14       1-1
12      1 1
10     11 0
8    21  1
6   12   1
4  11    0
2 12     1
0        0
-21       1
Integral Khovanov Homology

(db, data source)

  

Computer Talk

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

In[1]:=    
<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=
Crossings[Link[7, Alternating, 6]]
Out[2]=  
7
In[3]:=
PD[Link[7, Alternating, 6]]
Out[3]=  
PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[14, 10, 7, 9], X[12, 6, 13, 5], 
  X[2, 7, 3, 8], X[4, 12, 5, 11], X[6, 14, 1, 13]]
In[4]:=
GaussCode[Link[7, Alternating, 6]]
Out[4]=  
GaussCode[{1, -5, 2, -6, 4, -7}, {5, -1, 3, -2, 6, -4, 7, -3}]
In[5]:=
BR[Link[7, Alternating, 6]]
Out[5]=  
BR[Link[7, Alternating, 6]]
In[6]:=
alex = Alexander[Link[7, Alternating, 6]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[7, Alternating, 6]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[7, Alternating, 6]], KnotSignature[Link[7, Alternating, 6]]}
Out[9]=  
{Infinity, 3}
In[10]:=
J=Jones[Link[7, Alternating, 6]][q]
Out[10]=  
     1                    3/2      5/2      7/2      9/2      11/2

-(-------) + Sqrt[q] - 2 q + 2 q - 3 q + 2 q - 2 q +

 Sqrt[q]

  13/2
q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[7, Alternating, 6]][q]
Out[12]=  
     -2    2    4      8    10      12    14    20
1 + q   + q  + q  + 2 q  + q   + 2 q   + q   - q
In[13]:=
Kauffman[Link[7, Alternating, 6]][a, z]
Out[13]=  
                                           2    2    2    2      3
 -2    1      1    z    z    6 z   4 z   z    z    z    z    2 z

-a + ---- + --- + -- - -- - --- - --- - -- + -- + -- - -- - ---- +

       3     a z    7    5    3     a     8    6    4    2     7
      a  z         a    a    a           a    a    a    a     a

    3      3      3      4    4      4      5      5    5    6    6
 3 z    9 z    4 z    2 z    z    3 z    2 z    3 z    z    z    z
 ---- + ---- + ---- - ---- + -- + ---- - ---- - ---- - -- - -- - --
   5      3     a       6     4     2      5      3    a     4    2
a a a a a a a a a
In[14]:=
{Vassiliev[2][Link[7, Alternating, 6]], Vassiliev[3][Link[7, Alternating, 6]]}
Out[14]=  
      31

{0, -(--)}

48
In[15]:=
Kh[Link[7, Alternating, 6]][q, t]
Out[15]=  
                     2
  2    4     1     q     4      6        6  2      8  2    8  3

2 q + q + ----- + -- + q t + q t + 2 q t + 2 q t + q t +

            2  2   t
           q  t

  10  3    10  4    12  4    14  5
q t + q t + q t + q t