L8a5

From Knot Atlas
Revision as of 21:12, 28 August 2005 by ScottTestRobot (talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigationJump to search

L8a4.gif

L8a4

L8a6.gif

L8a6

L8a5.gif Visit L8a5's page at Knotilus!

Visit L8a5's page at the original Knot Atlas!

L8a5 is [math]\displaystyle{ 8^2_{11} }[/math] in the Rolfsen table of links.


L8a5 Further Notes and Views

Knot presentations

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

Polynomial invariants

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

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{17}{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:L8a5/V 2,1 Data:L8a5/V 3,1 Data:L8a5/V 4,1 Data:L8a5/V 4,2 Data:L8a5/V 4,3 Data:L8a5/V 5,1 Data:L8a5/V 5,2 Data:L8a5/V 5,3 Data:L8a5/V 5,4 Data:L8a5/V 6,1 Data:L8a5/V 6,2 Data:L8a5/V 6,3 Data:L8a5/V 6,4 Data:L8a5/V 6,5 Data:L8a5/V 6,6 Data:L8a5/V 6,7 Data:L8a5/V 6,8 Data:L8a5/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 L8a5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-101234χ
10        11
8       2 -2
6      21 1
4     32  -1
2    22   0
0   34    1
-2  11     0
-4  3      3
-611       0
-81        1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=4 }[/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[8, Alternating, 5]]
Out[2]=  
8
In[3]:=
PD[Link[8, Alternating, 5]]
Out[3]=  
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[14, 8, 15, 7], X[16, 10, 5, 9], 
  X[8, 16, 9, 15], X[10, 14, 11, 13], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[4]:=
GaussCode[Link[8, Alternating, 5]]
Out[4]=  
GaussCode[{1, -7, 2, -8}, {7, -1, 3, -5, 4, -6, 8, -2, 6, -3, 5, -4}]
In[5]:=
BR[Link[8, Alternating, 5]]
Out[5]=  
BR[Link[8, Alternating, 5]]
In[6]:=
alex = Alexander[Link[8, Alternating, 5]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[8, Alternating, 5]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[8, Alternating, 5]], KnotSignature[Link[8, Alternating, 5]]}
Out[9]=  
{Infinity, 1}
In[10]:=
J=Jones[Link[8, Alternating, 5]][q]
Out[10]=  
  -(7/2)    -(5/2)    4        4                     3/2      5/2

-q       + q       - ---- + ------- - 5 Sqrt[q] + 5 q    - 4 q    + 



                     3/2   Sqrt[q]
                    q

    7/2    9/2


  3 q    - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[8, Alternating, 5]][q]
Out[12]=  
     -12    2    2    4    2     -2      2      6    12    14

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



           10    8    6    4


           q     q    q    q
In[13]:=
Kauffman[Link[8, Alternating, 5]][a, z]
Out[13]=  
                                 3

     -2      2    1    3 a   2 a    z    2 z              3        2



-3 - a   - 3 a  + --- + --- + ---- - -- - --- - 6 a z - 5 a  z + 2 z  + 



                 a z    z     z      3    a
                                    a

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

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

  7
 z       7
 -- - a z


  a
In[14]:=
{Vassiliev[2][Link[8, Alternating, 5]], Vassiliev[3][Link[8, Alternating, 5]]}
Out[14]=  
      17

{0, -(--)}


      24
In[15]:=
Kh[Link[8, Alternating, 5]][q, t]
Out[15]=  
       2     1       1       1       3       1     3    1        2

4 + 2 q  + ----- + ----- + ----- + ----- + ----- + - + ---- + 2 q  t + 



           8  4    6  4    6  3    4  2    2  2   t    2
          q  t    q  t    q  t    q  t    q  t        q  t

    4        4  2      6  2    6  3      8  3    10  4


  3 q  t + 2 q  t  + 2 q  t  + q  t  + 2 q  t  + q   t
Retrieved from "https://katlas.org/index.php?title=L8a5&oldid=36982"