L9a12

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L9a11.gif

L9a11

L9a13.gif

L9a13

L9a12.gif Visit L9a12's page at Knotilus!

Visit L9a12's page at the original Knot Atlas!

L9a12 is [math]\displaystyle{ 9^2_{14} }[/math] in the Rolfsen table of links.


L9a12 Further Notes and Views

Knot presentations

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

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ \frac{-2 u v^4+2 u v^3-2 u v^2+2 u v-u-v^5+2 v^4-2 v^3+2 v^2-2 v}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ -\frac{4}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{23/2}}-\frac{2}{q^{21/2}}+\frac{4}{q^{19/2}}-\frac{5}{q^{17/2}}+\frac{6}{q^{15/2}}-\frac{7}{q^{13/2}}+\frac{4}{q^{11/2}} }[/math] (db)
Signature -5 (db)
HOMFLY-PT polynomial [math]\displaystyle{ a^{11} (-z)-2 a^{11} z^{-1} +3 a^9 z^3+9 a^9 z+5 a^9 z^{-1} -2 a^7 z^5-8 a^7 z^3-9 a^7 z-3 a^7 z^{-1} -a^5 z^5-3 a^5 z^3-a^5 z }[/math] (db)
Kauffman polynomial [math]\displaystyle{ -z^4 a^{14}+2 z^2 a^{14}-a^{14}-2 z^5 a^{13}+3 z^3 a^{13}-2 z^6 a^{12}+z^4 a^{12}+2 z^2 a^{12}-2 z^7 a^{11}+3 z^5 a^{11}-5 z^3 a^{11}+5 z a^{11}-2 a^{11} z^{-1} -z^8 a^{10}-z^6 a^{10}+6 z^4 a^{10}-11 z^2 a^{10}+5 a^{10}-5 z^7 a^9+17 z^5 a^9-26 z^3 a^9+16 z a^9-5 a^9 z^{-1} -z^8 a^8-z^6 a^8+9 z^4 a^8-12 z^2 a^8+5 a^8-3 z^7 a^7+11 z^5 a^7-15 z^3 a^7+10 z a^7-3 a^7 z^{-1} -2 z^6 a^6+5 z^4 a^6-z^2 a^6-z^5 a^5+3 z^3 a^5-z a^5 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ \frac{367}{24} }[/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:L9a12/V 2,1 Data:L9a12/V 3,1 Data:L9a12/V 4,1 Data:L9a12/V 4,2 Data:L9a12/V 4,3 Data:L9a12/V 5,1 Data:L9a12/V 5,2 Data:L9a12/V 5,3 Data:L9a12/V 5,4 Data:L9a12/V 6,1 Data:L9a12/V 6,2 Data:L9a12/V 6,3 Data:L9a12/V 6,4 Data:L9a12/V 6,5 Data:L9a12/V 6,6 Data:L9a12/V 6,7 Data:L9a12/V 6,8 Data:L9a12/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]-5 is the signature of L9a12. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -9-8-7-6-5-4-3-2-10χ
-4         11
-6        21-1
-8       2  2
-10      22  0
-12     52   3
-14    23    1
-16   34     -1
-18  12      1
-20 13       -2
-22 1        1
-241         -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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, 12]]
Out[2]=  
9
In[3]:=
PD[Link[9, Alternating, 12]]
Out[3]=  
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 13, 5, 14], X[14, 7, 15, 8], 

 X[16, 9, 17, 10], X[8, 15, 9, 16], X[10, 17, 11, 18], X[2, 5, 3, 6], 



  X[4, 11, 1, 12]]
In[4]:=
GaussCode[Link[9, Alternating, 12]]
Out[4]=  
GaussCode[{1, -8, 2, -9}, {8, -1, 4, -6, 5, -7, 9, -2, 3, -4, 6, -5, 7, 
   -3}]
In[5]:=
BR[Link[9, Alternating, 12]]
Out[5]=  
BR[Link[9, Alternating, 12]]
In[6]:=
alex = Alexander[Link[9, Alternating, 12]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[9, Alternating, 12]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[9, Alternating, 12]], KnotSignature[Link[9, Alternating, 12]]}
Out[9]=  
{Infinity, -5}
In[10]:=
J=Jones[Link[9, Alternating, 12]][q]
Out[10]=  
 -(23/2)     2       4       5       6       7       4      4

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



           21/2    19/2    17/2    15/2    13/2    11/2    9/2
          q       q       q       q       q       q       q

  2      -(5/2)
 ---- - q
  7/2


  q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[9, Alternating, 12]][q]
Out[12]=  
  -36    2     -32    2     -26    -24    4     2     4     2     -12

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



        34           30                  22    20    18    16
       q            q                   q     q     q     q

  -10    -8


  q    + q
In[13]:=
Kauffman[Link[9, Alternating, 12]][a, z]
Out[13]=  
                        7      9      11

  8      10    14   3 a    5 a    2 a      5         7         9



5 a  + 5 a   - a   - ---- - ---- - ----- - a  z + 10 a  z + 16 a  z + 



                     z      z       z

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

    5  3       7  3       9  3      11  3      13  3      6  4
 3 a  z  - 15 a  z  - 26 a  z  - 5 a   z  + 3 a   z  + 5 a  z  + 

    8  4      10  4    12  4    14  4    5  5       7  5       9  5
 9 a  z  + 6 a   z  + a   z  - a   z  - a  z  + 11 a  z  + 17 a  z  + 

    11  5      13  5      6  6    8  6    10  6      12  6      7  7
 3 a   z  - 2 a   z  - 2 a  z  - a  z  - a   z  - 2 a   z  - 3 a  z  - 

    9  7      11  7    8  8    10  8


  5 a  z  - 2 a   z  - a  z  - a   z
In[14]:=
{Vassiliev[2][Link[9, Alternating, 12]], Vassiliev[3][Link[9, Alternating, 12]]}
Out[14]=  
    367

{0, ---}


    24
In[15]:=
Kh[Link[9, Alternating, 12]][q, t]
Out[15]=  
 -6    -4     1        1        1        3        1        2

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



            24  9    22  8    20  8    20  7    18  7    18  6
           q   t    q   t    q   t    q   t    q   t    q   t

   3        4        2        3        5        2        2
 ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
  16  6    16  5    14  5    14  4    12  4    12  3    10  3
 q   t    q   t    q   t    q   t    q   t    q   t    q   t

   2        2      2
 ------ + ----- + ----
  10  2    8  2    6


  q   t    q  t    q  t
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