T(8,3)

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[[Image:T(15,2).{{{ext}}}|80px|link=T(15,2)]]

T(15,2)

[[Image:T(17,2).{{{ext}}}|80px|link=T(17,2)]]

T(17,2)

T(8,3).jpg Visit T(8,3)'s page at Knotilus!

Visit T(8,3)'s page at the original Knot Atlas!

T(8,3) Quick Notes



Banco Internacional do Funchal [1]

Knot presentations

Planar diagram presentation X11,1,12,32 X22,2,23,1 X23,13,24,12 X2,14,3,13 X3,25,4,24 X14,26,15,25 X15,5,16,4 X26,6,27,5 X27,17,28,16 X6,18,7,17 X7,29,8,28 X18,30,19,29 X19,9,20,8 X30,10,31,9 X31,21,32,20 X10,22,11,21
Gauss code {2, -4, -5, 7, 8, -10, -11, 13, 14, -16, -1, 3, 4, -6, -7, 9, 10, -12, -13, 15, 16, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 1}
Dowker-Thistlethwaite code 22 -24 26 -28 30 -32 2 -4 6 -8 10 -12 14 -16 18 -20

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^7-t^6+t^4-t^3+t-1+ t^{-1} - t^{-3} + t^{-4} - t^{-6} + t^{-7} }[/math]
Conway polynomial [math]\displaystyle{ z^{14}+13 z^{12}+65 z^{10}+157 z^8+189 z^6+105 z^4+21 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 3, 10 }
Jones polynomial [math]\displaystyle{ -q^{16}+q^9+q^7 }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^{14} a^{-14} +14 z^{12} a^{-14} -z^{12} a^{-16} +78 z^{10} a^{-14} -13 z^{10} a^{-16} +221 z^8 a^{-14} -65 z^8 a^{-16} +z^8 a^{-18} +338 z^6 a^{-14} -157 z^6 a^{-16} +8 z^6 a^{-18} +273 z^4 a^{-14} -189 z^4 a^{-16} +21 z^4 a^{-18} +105 z^2 a^{-14} -105 z^2 a^{-16} +21 z^2 a^{-18} +15 a^{-14} -21 a^{-16} +7 a^{-18} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^{14} a^{-14} +z^{14} a^{-16} +z^{13} a^{-15} +z^{13} a^{-17} -14 z^{12} a^{-14} -14 z^{12} a^{-16} -13 z^{11} a^{-15} -13 z^{11} a^{-17} +78 z^{10} a^{-14} +78 z^{10} a^{-16} +65 z^9 a^{-15} +65 z^9 a^{-17} -221 z^8 a^{-14} -222 z^8 a^{-16} -z^8 a^{-18} -157 z^7 a^{-15} -157 z^7 a^{-17} +338 z^6 a^{-14} +346 z^6 a^{-16} +8 z^6 a^{-18} +189 z^5 a^{-15} +189 z^5 a^{-17} -273 z^4 a^{-14} -294 z^4 a^{-16} -21 z^4 a^{-18} -105 z^3 a^{-15} -105 z^3 a^{-17} +105 z^2 a^{-14} +126 z^2 a^{-16} +21 z^2 a^{-18} +21 z a^{-15} +21 z a^{-17} -15 a^{-14} -21 a^{-16} -7 a^{-18} }[/math]
The A2 invariant Data:T(8,3)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(8,3)/QuantumInvariant/G2/1,0
Further Quantum Invariants
T(8,3) Quantum Invariants.
Computer Talk
The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["T(8,3)"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^7-t^6+t^4-t^3+t-1+ t^{-1} - t^{-3} + t^{-4} - t^{-6} + t^{-7} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^{14}+13 z^{12}+65 z^{10}+157 z^8+189 z^6+105 z^4+21 z^2+1 }[/math]
In[6]:= Alexander[K, 2][t]
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
Out[6]= [math]\displaystyle{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 3, 10 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^{16}+q^9+q^7 }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^{14} a^{-14} +14 z^{12} a^{-14} -z^{12} a^{-16} +78 z^{10} a^{-14} -13 z^{10} a^{-16} +221 z^8 a^{-14} -65 z^8 a^{-16} +z^8 a^{-18} +338 z^6 a^{-14} -157 z^6 a^{-16} +8 z^6 a^{-18} +273 z^4 a^{-14} -189 z^4 a^{-16} +21 z^4 a^{-18} +105 z^2 a^{-14} -105 z^2 a^{-16} +21 z^2 a^{-18} +15 a^{-14} -21 a^{-16} +7 a^{-18} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^{14} a^{-14} +z^{14} a^{-16} +z^{13} a^{-15} +z^{13} a^{-17} -14 z^{12} a^{-14} -14 z^{12} a^{-16} -13 z^{11} a^{-15} -13 z^{11} a^{-17} +78 z^{10} a^{-14} +78 z^{10} a^{-16} +65 z^9 a^{-15} +65 z^9 a^{-17} -221 z^8 a^{-14} -222 z^8 a^{-16} -z^8 a^{-18} -157 z^7 a^{-15} -157 z^7 a^{-17} +338 z^6 a^{-14} +346 z^6 a^{-16} +8 z^6 a^{-18} +189 z^5 a^{-15} +189 z^5 a^{-17} -273 z^4 a^{-14} -294 z^4 a^{-16} -21 z^4 a^{-18} -105 z^3 a^{-15} -105 z^3 a^{-17} +105 z^2 a^{-14} +126 z^2 a^{-16} +21 z^2 a^{-18} +21 z a^{-15} +21 z a^{-17} -15 a^{-14} -21 a^{-16} -7 a^{-18} }[/math]

Vassiliev invariants

V2 and V3 {0, 84}

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]10 is the signature of T(8,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 01234567891011χ
33           1-1
31         1  -1
29         11 0
27       11   0
25     1  1   0
23     11     0
21   11       0
19    1       1
17  1         1
151           1
131           1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Include[ColouredJonesM.mhtml] 

In[1]:=    
 
<< KnotTheory`
 
Loading KnotTheory` (version of August 19, 2005, 13:11:25)...
In[2]:=
Crossings[TorusKnot[8, 3]]
Out[2]=  
16
In[3]:=
PD[TorusKnot[8, 3]]
Out[3]=  
PD[X[11, 1, 12, 32], X[22, 2, 23, 1], X[23, 13, 24, 12], 

 X[2, 14, 3, 13], X[3, 25, 4, 24], X[14, 26, 15, 25], X[15, 5, 16, 4], 

 X[26, 6, 27, 5], X[27, 17, 28, 16], X[6, 18, 7, 17], X[7, 29, 8, 28], 

 X[18, 30, 19, 29], X[19, 9, 20, 8], X[30, 10, 31, 9], 



  X[31, 21, 32, 20], X[10, 22, 11, 21]]
In[4]:=
GaussCode[TorusKnot[8, 3]]
Out[4]=  
GaussCode[2, -4, -5, 7, 8, -10, -11, 13, 14, -16, -1, 3, 4, -6, -7, 9, 
  10, -12, -13, 15, 16, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 1]
In[5]:=
BR[TorusKnot[8, 3]]
Out[5]=  
BR[3, {1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2}]
In[6]:=
alex = Alexander[TorusKnot[8, 3]][t]
Out[6]=  
      -7    -6    -4    -3   1        3    4    6    7

-1 + t   - t   + t   - t   + - + t - t  + t  - t  + t


                             t
In[7]:=
Conway[TorusKnot[8, 3]][z]
Out[7]=  
        2        4        6        8       10       12    14
1 + 21 z  + 105 z  + 189 z  + 157 z  + 65 z   + 13 z   + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[TorusKnot[8, 3]], KnotSignature[TorusKnot[8, 3]]}
Out[9]=  
{3, 10}
In[10]:=
J=Jones[TorusKnot[8, 3]][q]
Out[10]=  
 7    9    16
q  + q  - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[TorusKnot[8, 3]][q]
Out[12]=  
NotAvailable
In[13]:=
Kauffman[TorusKnot[8, 3]][a, z]
Out[13]=  
                                    2        2        2        3

-7    21    15    21 z   21 z   21 z    126 z    105 z    105 z
--- - --- - --- + ---- + ---- + ----- + ------ + ------ - ------ - 



18    16    14    17     15      18      16       14       17



a     a     a     a      a       a       a        a        a



      3       4        4        4        5        5      6        6
 105 z    21 z    294 z    273 z    189 z    189 z    8 z    346 z
 ------ - ----- - ------ - ------ + ------ + ------ + ---- + ------ + 
   15       18      16       14       17       15      18      16
  a        a       a        a        a        a       a       a

      6        7        7    8         8        8       9       9
 338 z    157 z    157 z    z     222 z    221 z    65 z    65 z
 ------ - ------ - ------ - --- - ------ - ------ + ----- + ----- + 
   14       17       15      18     16       14       17      15
  a        a        a       a      a        a        a       a

     10       10       11       11       12       12    13    13
 78 z     78 z     13 z     13 z     14 z     14 z     z     z
 ------ + ------ - ------ - ------ - ------ - ------ + --- + --- + 
   16       14       17       15       16       14      17    15
  a        a        a        a        a        a       a     a

  14    14
 z     z
 --- + ---
  16    14


  a     a
In[14]:=
{Vassiliev[2][TorusKnot[8, 3]], Vassiliev[3][TorusKnot[8, 3]]}
Out[14]=  
{0, 84}
In[15]:=
Kh[TorusKnot[8, 3]][q, t]
Out[15]=  
 13    15    17  2    21  3    19  4    21  4    23  5    25  5

q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 



  23  6    27  7    25  8    27  8    29  9    31  9    29  10
 q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t   + 

  33  11


  q   t
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