L9a47

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L9a46.gif

L9a46

L9a48.gif

L9a48

L9a47.gif Visit L9a47's page at Knotilus!

Visit L9a47's page at the original Knot Atlas!

L9a47 is [math]\displaystyle{ 9^3_{2} }[/math] in the Rolfsen table of links.


L9a47 Further Notes and Views

Knot presentations

Planar diagram presentation X6172 X12,3,13,4 X10,13,5,14 X18,15,11,16 X14,7,15,8 X8,18,9,17 X16,10,17,9 X2536 X4,11,1,12
Gauss code {1, -8, 2, -9}, {8, -1, 5, -6, 7, -3}, {9, -2, 3, -5, 4, -7, 6, -4}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ \frac{-t(3)^2 t(2)^2+t(1) t(2)^2-t(1) t(3) t(2)^2+2 t(3) t(2)^2-t(2)^2-t(1) t(3)^2 t(2)+2 t(3)^2 t(2)-2 t(1) t(2)+4 t(1) t(3) t(2)-4 t(3) t(2)+t(2)+t(1) t(3)^2-t(3)^2+t(1)-2 t(1) t(3)+t(3)}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial [math]\displaystyle{ - q^{-8} +3 q^{-7} -5 q^{-6} +8 q^{-5} -8 q^{-4} +10 q^{-3} -7 q^{-2} +q+6 q^{-1} -3 }[/math] (db)
Signature -2 (db)
HOMFLY-PT polynomial [math]\displaystyle{ -a^8+3 z^2 a^6+a^6 z^{-2} +3 a^6-2 z^4 a^4-4 z^2 a^4-2 a^4 z^{-2} -5 a^4-z^4 a^2+z^2 a^2+a^2 z^{-2} +3 a^2+z^2 }[/math] (db)
Kauffman polynomial [math]\displaystyle{ z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-7 z^4 a^8+5 z^2 a^8-2 a^8+3 z^7 a^7-3 z^5 a^7-4 z^3 a^7+3 z a^7+z^8 a^6+8 z^6 a^6-24 z^4 a^6+21 z^2 a^6+a^6 z^{-2} -8 a^6+7 z^7 a^5-10 z^5 a^5+5 z a^5-2 a^5 z^{-1} +z^8 a^4+10 z^6 a^4-26 z^4 a^4+24 z^2 a^4+2 a^4 z^{-2} -9 a^4+4 z^7 a^3-3 z^5 a^3-z^3 a^3+3 z a^3-2 a^3 z^{-1} +5 z^6 a^2-8 z^4 a^2+7 z^2 a^2+a^2 z^{-2} -4 a^2+3 z^5 a-3 z^3 a+z^4-z^2 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ \frac{11}{3} }[/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:L9a47/V 2,1 Data:L9a47/V 3,1 Data:L9a47/V 4,1 Data:L9a47/V 4,2 Data:L9a47/V 4,3 Data:L9a47/V 5,1 Data:L9a47/V 5,2 Data:L9a47/V 5,3 Data:L9a47/V 5,4 Data:L9a47/V 6,1 Data:L9a47/V 6,2 Data:L9a47/V 6,3 Data:L9a47/V 6,4 Data:L9a47/V 6,5 Data:L9a47/V 6,6 Data:L9a47/V 6,7 Data:L9a47/V 6,8 Data:L9a47/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]-2 is the signature of L9a47. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -7-6-5-4-3-2-1012χ
3         11
1        2 -2
-1       41 3
-3      43  -1
-5     63   3
-7    46    2
-9   44     0
-11  25      3
-13 13       -2
-15 2        2
-171         -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math]
[math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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, 47]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 47]]
Out[3]=  
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[10, 13, 5, 14], X[18, 15, 11, 16], 

 X[14, 7, 15, 8], X[8, 18, 9, 17], X[16, 10, 17, 9], X[2, 5, 3, 6], 



  X[4, 11, 1, 12]]
In[4]:=
GaussCode[Link[9, Alternating, 47]]
Out[4]=  
GaussCode[{1, -8, 2, -9}, {8, -1, 5, -6, 7, -3}, 
  {9, -2, 3, -5, 4, -7, 6, -4}]
In[5]:=
BR[Link[9, Alternating, 47]]
Out[5]=  
BR[Link[9, Alternating, 47]]
In[6]:=
alex = Alexander[Link[9, Alternating, 47]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 47]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 47]], KnotSignature[Link[9, Alternating, 47]]}
Out[9]=  
{Infinity, -2}
In[10]:=
J=Jones[Link[9, Alternating, 47]][q]
Out[10]=  
      -8   3    5    8    8    10   7    6

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



           7    6    5    4    3    2   q


           q    q    q    q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 47]][q]
Out[12]=  
      -26    -24    2     2     5     2     5     3    4    4    3

-1 - q    - q    + --- + --- + --- + --- + --- + --- + -- + -- + -- - 



                   22    18    16    14    12    10    8    6    2
                  q     q     q     q     q     q     q    q    q

  2    4


  q  + q
In[13]:=
Kauffman[Link[9, Alternating, 47]][a, z]
Out[13]=  
                              2      4    6      3      5

   2      4      6      8   a    2 a    a    2 a    2 a       3



-4 a  - 9 a  - 8 a  - 2 a  + -- + ---- + -- - ---- - ---- + 3 a  z + 



                             2     2     2    z      z
                            z     z     z

    5        7      9      2      2  2       4  2       6  2
 5 a  z + 3 a  z + a  z - z  + 7 a  z  + 24 a  z  + 21 a  z  + 

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

     4  4       6  4      8  4        5      3  5       5  5
 26 a  z  - 24 a  z  - 7 a  z  + 3 a z  - 3 a  z  - 10 a  z  - 

    7  5    9  5      2  6       4  6      6  6      8  6      3  7
 3 a  z  + a  z  + 5 a  z  + 10 a  z  + 8 a  z  + 3 a  z  + 4 a  z  + 

    5  7      7  7    4  8    6  8


  7 a  z  + 3 a  z  + a  z  + a  z
In[14]:=
{Vassiliev[2][Link[9, Alternating, 47]], Vassiliev[3][Link[9, Alternating, 47]]}
Out[14]=  
    11

{0, --}


    3
In[15]:=
Kh[Link[9, Alternating, 47]][q, t]
Out[15]=  
3    4     1        2        1        3        2        5        4

-- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 



3   q    17  7    15  6    13  6    13  5    11  5    11  4    9  4



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



   4       4       6       6      3      4     t            3  2
 ----- + ----- + ----- + ----- + ---- + ---- + - + 2 q t + q  t
  9  3    7  3    7  2    5  2    5      3     q


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