L9a43

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L9a42.gif

L9a42

L9a44.gif

L9a44

L9a43.gif Visit L9a43's page at Knotilus!

Visit L9a43's page at the original Knot Atlas!

L9a43 is [math]\displaystyle{ 9^3_{4} }[/math] in the Rolfsen table of links.


L9a43 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ \frac{t(1) t(3)^3+t(2) t(3)^3-t(3)^3-3 t(1) t(3)^2+2 t(1) t(2) t(3)^2-3 t(2) t(3)^2+2 t(3)^2+3 t(1) t(3)-2 t(1) t(2) t(3)+3 t(2) t(3)-2 t(3)-t(1)+t(1) t(2)-t(2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ - q^{-11} +2 q^{-10} -5 q^{-9} +8 q^{-8} -8 q^{-7} +10 q^{-6} -7 q^{-5} +7 q^{-4} -3 q^{-3} + q^{-2} }[/math] (db)
Signature -4 (db)
HOMFLY-PT polynomial [math]\displaystyle{ -a^{12} z^{-2} +4 a^{10} z^{-2} +4 a^{10}-6 a^8 z^2-5 a^8 z^{-2} -11 a^8+3 a^6 z^4+8 a^6 z^2+2 a^6 z^{-2} +7 a^6+a^4 z^4+a^4 z^2 }[/math] (db)
Kauffman polynomial [math]\displaystyle{ a^{13} z^5-3 a^{13} z^3+3 a^{13} z-a^{13} z^{-1} +2 a^{12} z^6-4 a^{12} z^4+3 a^{12} z^2+a^{12} z^{-2} -2 a^{12}+2 a^{11} z^7+a^{11} z^5-11 a^{11} z^3+13 a^{11} z-5 a^{11} z^{-1} +a^{10} z^8+6 a^{10} z^6-16 a^{10} z^4+14 a^{10} z^2+4 a^{10} z^{-2} -10 a^{10}+6 a^9 z^7-3 a^9 z^5-16 a^9 z^3+21 a^9 z-9 a^9 z^{-1} +a^8 z^8+10 a^8 z^6-24 a^8 z^4+23 a^8 z^2+5 a^8 z^{-2} -14 a^8+4 a^7 z^7-10 a^7 z^3+11 a^7 z-5 a^7 z^{-1} +6 a^6 z^6-11 a^6 z^4+11 a^6 z^2+2 a^6 z^{-2} -7 a^6+3 a^5 z^5-2 a^5 z^3+a^4 z^4-a^4 z^2 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ \frac{1}{2} }[/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:L9a43/V 2,1 Data:L9a43/V 3,1 Data:L9a43/V 4,1 Data:L9a43/V 4,2 Data:L9a43/V 4,3 Data:L9a43/V 5,1 Data:L9a43/V 5,2 Data:L9a43/V 5,3 Data:L9a43/V 5,4 Data:L9a43/V 6,1 Data:L9a43/V 6,2 Data:L9a43/V 6,3 Data:L9a43/V 6,4 Data:L9a43/V 6,5 Data:L9a43/V 6,6 Data:L9a43/V 6,7 Data:L9a43/V 6,8 Data:L9a43/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]-4 is the signature of L9a43. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -9-8-7-6-5-4-3-2-10χ
-3         11
-5        31-2
-7       4  4
-9      33  0
-11     74   3
-13    46    2
-15   44     0
-17  14      3
-19 14       -3
-21 1        1
-231         -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math]
[math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=0 }[/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[9, Alternating, 43]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 43]]
Out[3]=  
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 7, 15, 8], X[8, 13, 5, 14], 

 X[16, 11, 17, 12], X[18, 15, 9, 16], X[12, 17, 13, 18], 



  X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[4]:=
GaussCode[Link[9, Alternating, 43]]
Out[4]=  
GaussCode[{1, -8, 2, -9}, {8, -1, 3, -4}, 
  {9, -2, 5, -7, 4, -3, 6, -5, 7, -6}]
In[5]:=
BR[Link[9, Alternating, 43]]
Out[5]=  
BR[Link[9, Alternating, 43]]
In[6]:=
alex = Alexander[Link[9, Alternating, 43]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 43]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 43]], KnotSignature[Link[9, Alternating, 43]]}
Out[9]=  
{Infinity, -4}
In[10]:=
J=Jones[Link[9, Alternating, 43]][q]
Out[10]=  
  -11    2    5    8    8    10   7    7    3     -2

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



        10    9    8    7    6    5    4    3


        q     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, 43]][q]
Out[12]=  
  -36    3     2     -30    2     4     5     6     8     4     6

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



        34    32           28    26    24    22    20    18    16
       q     q            q     q     q     q     q     q     q

  -14    -12    2    2     -6
 q    + q    + --- - -- + q
                10    8


                q     q
In[13]:=
Kauffman[Link[9, Alternating, 43]][a, z]
Out[13]=  
                                    6      8      10    12      7

   6       8       10      12   2 a    5 a    4 a     a     5 a



-7 a  - 14 a  - 10 a   - 2 a   + ---- + ---- + ----- + --- - ---- - 



                                  2      2      2      2     z
                                 z      z      z      z

    9      11    13
 9 a    5 a     a         7         9         11        13      4  2
 ---- - ----- - --- + 11 a  z + 21 a  z + 13 a   z + 3 a   z - a  z  + 
  z       z      z

     6  2       8  2       10  2      12  2      5  3       7  3
 11 a  z  + 23 a  z  + 14 a   z  + 3 a   z  - 2 a  z  - 10 a  z  - 

     9  3       11  3      13  3    4  4       6  4       8  4
 16 a  z  - 11 a   z  - 3 a   z  + a  z  - 11 a  z  - 24 a  z  - 

     10  4      12  4      5  5      9  5    11  5    13  5
 16 a   z  - 4 a   z  + 3 a  z  - 3 a  z  + a   z  + a   z  + 

    6  6       8  6      10  6      12  6      7  7      9  7
 6 a  z  + 10 a  z  + 6 a   z  + 2 a   z  + 4 a  z  + 6 a  z  + 

    11  7    8  8    10  8


  2 a   z  + a  z  + a   z
In[14]:=
{Vassiliev[2][Link[9, Alternating, 43]], Vassiliev[3][Link[9, Alternating, 43]]}
Out[14]=  
    1

{0, -}


    2
In[15]:=
Kh[Link[9, Alternating, 43]][q, t]
Out[15]=  
 -5    -3     1        1        1        4        1        4

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



            23  9    21  8    19  8    19  7    17  7    17  6
           q   t    q   t    q   t    q   t    q   t    q   t

   4        4        4        6        7        4        3       3
 ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 
  15  6    15  5    13  5    13  4    11  4    11  3    9  3    9  2
 q   t    q   t    q   t    q   t    q   t    q   t    q  t    q  t

   4      3
 ----- + ----
  7  2    5


  q  t    q  t
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