L9a25

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L9a24.gif

L9a24

L9a26.gif

L9a26

L9a25.gif Visit L9a25's page at Knotilus!

Visit L9a25's page at the original Knot Atlas!

L9a25 is [math]\displaystyle{ 9^2_{8} }[/math] in the Rolfsen table of links.


L9a25 Further Notes and Views

Knot presentations

Planar diagram presentation X8192 X10,3,11,4 X14,6,15,5 X18,11,7,12 X16,13,17,14 X12,17,13,18 X4,16,5,15 X2738 X6,9,1,10
Gauss code {1, -8, 2, -7, 3, -9}, {8, -1, 9, -2, 4, -6, 5, -3, 7, -5, 6, -4}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ \frac{2 t(2) t(1)^2-2 t(1)^2+2 t(2)^2 t(1)-5 t(2) t(1)+2 t(1)-2 t(2)^2+2 t(2)}{t(1) t(2)} }[/math] (db)
Jones polynomial [math]\displaystyle{ -\frac{5}{q^{9/2}}+\frac{5}{q^{7/2}}-\frac{6}{q^{5/2}}-q^{3/2}+\frac{5}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{3}{q^{11/2}}+2 \sqrt{q}-\frac{4}{\sqrt{q}} }[/math] (db)
Signature -1 (db)
HOMFLY-PT polynomial [math]\displaystyle{ -z a^7+z^3 a^5+2 z^3 a^3+2 z a^3+a^3 z^{-1} +z^3 a-z a-a z^{-1} -z a^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ -z^6 a^8+4 z^4 a^8-4 z^2 a^8-2 z^7 a^7+8 z^5 a^7-9 z^3 a^7+3 z a^7-z^8 a^6+z^6 a^6+4 z^4 a^6-3 z^2 a^6-4 z^7 a^5+11 z^5 a^5-7 z^3 a^5+2 z a^5-z^8 a^4-z^6 a^4+5 z^4 a^4-z^2 a^4-2 z^7 a^3+6 z^3 a^3-5 z a^3+a^3 z^{-1} -3 z^6 a^2+3 z^4 a^2-z^2 a^2-a^2-3 z^5 a+3 z^3 a-3 z a+a z^{-1} -2 z^4+z^2-z^3 a^{-1} +z a^{-1} }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ \frac{79}{48} }[/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:L9a25/V 2,1 Data:L9a25/V 3,1 Data:L9a25/V 4,1 Data:L9a25/V 4,2 Data:L9a25/V 4,3 Data:L9a25/V 5,1 Data:L9a25/V 5,2 Data:L9a25/V 5,3 Data:L9a25/V 5,4 Data:L9a25/V 6,1 Data:L9a25/V 6,2 Data:L9a25/V 6,3 Data:L9a25/V 6,4 Data:L9a25/V 6,5 Data:L9a25/V 6,6 Data:L9a25/V 6,7 Data:L9a25/V 6,8 Data:L9a25/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]-1 is the signature of L9a25. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -7-6-5-4-3-2-1012χ
4         11
2        1 -1
0       31 2
-2      32  -1
-4     32   1
-6    34    1
-8   22     0
-10  13      2
-12 12       -1
-14 1        1
-161         -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math]
[math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/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[9, Alternating, 25]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 25]]
Out[3]=  
PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[14, 6, 15, 5], X[18, 11, 7, 12], 

 X[16, 13, 17, 14], X[12, 17, 13, 18], X[4, 16, 5, 15], X[2, 7, 3, 8], 



  X[6, 9, 1, 10]]
In[4]:=
GaussCode[Link[9, Alternating, 25]]
Out[4]=  
GaussCode[{1, -8, 2, -7, 3, -9}, 
  {8, -1, 9, -2, 4, -6, 5, -3, 7, -5, 6, -4}]
In[5]:=
BR[Link[9, Alternating, 25]]
Out[5]=  
BR[Link[9, Alternating, 25]]
In[6]:=
alex = Alexander[Link[9, Alternating, 25]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 25]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 25]], KnotSignature[Link[9, Alternating, 25]]}
Out[9]=  
{Infinity, -1}
In[10]:=
J=Jones[Link[9, Alternating, 25]][q]
Out[10]=  
 -(15/2)     2       3      5      5      6      5        4

q        - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 



           13/2    11/2    9/2    7/2    5/2    3/2   Sqrt[q]
          q       q       q      q      q      q

              3/2


  2 Sqrt[q] - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 25]][q]
Out[12]=  
     -24    -20    -18    -16    -14    2     -8   2     -4    2    6

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



                                       10          6


                                       q           q
In[13]:=
Kauffman[Link[9, Alternating, 25]][a, z]
Out[13]=  
           3

 2   a   a    z              3        5        7      2    2  2



-a  + - + -- + - - 3 a z - 5 a  z + 2 a  z + 3 a  z + z  - a  z  - 



     z   z    a

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

    7  3      4      2  4      4  4      6  4      8  4        5
 9 a  z  - 2 z  + 3 a  z  + 5 a  z  + 4 a  z  + 4 a  z  - 3 a z  + 

     5  5      7  5      2  6    4  6    6  6    8  6      3  7
 11 a  z  + 8 a  z  - 3 a  z  - a  z  + a  z  - a  z  - 2 a  z  - 

    5  7      7  7    4  8    6  8


  4 a  z  - 2 a  z  - a  z  - a  z
In[14]:=
{Vassiliev[2][Link[9, Alternating, 25]], Vassiliev[3][Link[9, Alternating, 25]]}
Out[14]=  
    79

{0, --}


    48
In[15]:=
Kh[Link[9, Alternating, 25]][q, t]
Out[15]=  
    2      1        1        1        2        1        3        2

3 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 



    2    16  7    14  6    12  6    12  5    10  5    10  4    8  4
   q    q   t    q   t    q   t    q   t    q   t    q   t    q  t

   2       3       4       3      2      3          2      4  2
 ----- + ----- + ----- + ----- + ---- + ---- + t + q  t + q  t
  8  3    6  3    6  2    4  2    4      2


  q  t    q  t    q  t    q  t    q  t   q  t
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