© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(13,3)
T(13,3)
T(27,2)
T(27,2)
T(9,4)
TubePlot
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   The 27-Crossing Torus Knot T(9,4)

Visit T(9,4)'s page at Knotilus!

Acknowledgement

PD Presentation: X11,25,12,24 X52,26,53,25 X39,27,40,26 X53,13,54,12 X40,14,41,13 X27,15,28,14 X41,1,42,54 X28,2,29,1 X15,3,16,2 X29,43,30,42 X16,44,17,43 X3,45,4,44 X17,31,18,30 X4,32,5,31 X45,33,46,32 X5,19,6,18 X46,20,47,19 X33,21,34,20 X47,7,48,6 X34,8,35,7 X21,9,22,8 X35,49,36,48 X22,50,23,49 X9,51,10,50 X23,37,24,36 X10,38,11,37 X51,39,52,38

Gauss Code: {8, 9, -12, -14, -16, 19, 20, 21, -24, -26, -1, 4, 5, 6, -9, -11, -13, 16, 17, 18, -21, -23, -25, 1, 2, 3, -6, -8, -10, 13, 14, 15, -18, -20, -22, 25, 26, 27, -3, -5, -7, 10, 11, 12, -15, -17, -19, 22, 23, 24, -27, -2, -4, 7}

Braid Representative:    

Alexander Polynomial: t-12 - t-11 + t-8 - t-7 + t-4 - t-2 + 1 - t2 + t4 - t7 + t8 - t11 + t12

Conway Polynomial: 1 + 50z2 + 665z4 + 3675z6 + 10318z8 + 16720z10 + 16834z12 + 10963z14 + 4693z16 + 1311z18 + 230z20 + 23z22 + z24

Other knots with the same Alexander/Conway Polynomial: {...}

Determinant and Signature: {9, 16}

Jones Polynomial: q12 + q14 + q16 - q17 + q18 - q19 + q20 - q21 - q23

Other knots (up to mirrors) with the same Jones Polynomial: {...}

A2 (sl(3)) Invariant: Not Available.

Kauffman Polynomial: Not Available.

V2 and V3, the type 2 and 3 Vassiliev invariants: {50, 300}

Khovanov Homology. The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=16 is the signature of T(9,4). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
01234567891011121314151617χ
51                110
49                  0
47               21 -1
45            12    -1
43           1 11   -1
41           22     0
39         21 1     0
37       1  1       0
35     1 12         0
33     11           0
31   11 1           1
29    1             1
27  1               1
251                 1
231                 1


Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=    
<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=
TubePlot[TorusKnot[9, 4]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[9, 4]]
Out[3]=   
27
In[4]:=
PD[TorusKnot[9, 4]]
Out[4]=   
PD[X[11, 25, 12, 24], X[52, 26, 53, 25], X[39, 27, 40, 26], X[53, 13, 54, 12], 
 
>   X[40, 14, 41, 13], X[27, 15, 28, 14], X[41, 1, 42, 54], X[28, 2, 29, 1], 
 
>   X[15, 3, 16, 2], X[29, 43, 30, 42], X[16, 44, 17, 43], X[3, 45, 4, 44], 
 
>   X[17, 31, 18, 30], X[4, 32, 5, 31], X[45, 33, 46, 32], X[5, 19, 6, 18], 
 
>   X[46, 20, 47, 19], X[33, 21, 34, 20], X[47, 7, 48, 6], X[34, 8, 35, 7], 
 
>   X[21, 9, 22, 8], X[35, 49, 36, 48], X[22, 50, 23, 49], X[9, 51, 10, 50], 
 
>   X[23, 37, 24, 36], X[10, 38, 11, 37], X[51, 39, 52, 38]]
In[5]:=
GaussCode[TorusKnot[9, 4]]
Out[5]=   
GaussCode[8, 9, -12, -14, -16, 19, 20, 21, -24, -26, -1, 4, 5, 6, -9, -11, -13, 
 
>   16, 17, 18, -21, -23, -25, 1, 2, 3, -6, -8, -10, 13, 14, 15, -18, -20, -22, 
 
>   25, 26, 27, -3, -5, -7, 10, 11, 12, -15, -17, -19, 22, 23, 24, -27, -2, -4, 
 
>   7]
In[6]:=
BR[TorusKnot[9, 4]]
Out[6]=   
BR[4, {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 
 
>    1, 2, 3}]
In[7]:=
alex = Alexander[TorusKnot[9, 4]][t]
Out[7]=   
     -12    -11    -8    -7    -4    -2    2    4    7    8    11    12
1 + t    - t    + t   - t   + t   - t   - t  + t  - t  + t  - t   + t
In[8]:=
Conway[TorusKnot[9, 4]][z]
Out[8]=   
        2        4         6          8          10          12          14
1 + 50 z  + 665 z  + 3675 z  + 10318 z  + 16720 z   + 16834 z   + 10963 z   + 
 
          16         18        20       22    24
>   4693 z   + 1311 z   + 230 z   + 23 z   + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{}
In[10]:=
{KnotDet[TorusKnot[9, 4]], KnotSignature[TorusKnot[9, 4]]}
Out[10]=   
{9, 16}
In[11]:=
J=Jones[TorusKnot[9, 4]][q]
Out[11]=   
 12    14    16    17    18    19    20    21    23
q   + q   + q   - q   + q   - q   + q   - q   - q
In[12]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[12]=   
{}
In[13]:=
A2Invariant[TorusKnot[9, 4]][q]
Out[13]=   
NotAvailable
In[14]:=
Kauffman[TorusKnot[9, 4]][a, z]
Out[14]=   
NotAvailable
In[15]:=
{Vassiliev[2][TorusKnot[9, 4]], Vassiliev[3][TorusKnot[9, 4]]}
Out[15]=   
{50, 300}
In[16]:=
Kh[TorusKnot[9, 4]][q, t]
Out[16]=   
 23    25    27  2    31  3    29  4    31  4    33  5    35  5    31  6
q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 
 
     33  6    35  7    37  7      35  8      39  9    37  10    39  10
>   q   t  + q   t  + q   t  + 2 q   t  + 2 q   t  + q   t   + q   t   + 
 
       41  11    43  11    39  12      41  12    45  12    43  13      45  13
>   2 q   t   + q   t   + q   t   + 2 q   t   + q   t   + q   t   + 2 q   t   + 
 
     43  14      47  15    47  16    51  16    51  17
>   q   t   + 2 q   t   + q   t   + q   t   + q   t


Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(9,4)
T(13,3)
T(13,3)
T(27,2)
T(27,2)