© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(11,4)
T(11,4)
T(17,3)
T(17,3)
T(33,2)
TubePlot
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   The 33-Crossing Torus Knot T(33,2)

Visit T(33,2)'s page at Knotilus!

Acknowledgement

PD Presentation: X31,65,32,64 X65,33,66,32 X33,1,34,66 X1,35,2,34 X35,3,36,2 X3,37,4,36 X37,5,38,4 X5,39,6,38 X39,7,40,6 X7,41,8,40 X41,9,42,8 X9,43,10,42 X43,11,44,10 X11,45,12,44 X45,13,46,12 X13,47,14,46 X47,15,48,14 X15,49,16,48 X49,17,50,16 X17,51,18,50 X51,19,52,18 X19,53,20,52 X53,21,54,20 X21,55,22,54 X55,23,56,22 X23,57,24,56 X57,25,58,24 X25,59,26,58 X59,27,60,26 X27,61,28,60 X61,29,62,28 X29,63,30,62 X63,31,64,30

Gauss Code: {-4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -28, 29, -30, 31, -32, 33, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 32, -33, 1, -2, 3}

Braid Representative:    

Alexander Polynomial: t-16 - t-15 + t-14 - t-13 + t-12 - t-11 + t-10 - t-9 + t-8 - t-7 + t-6 - t-5 + t-4 - t-3 + t-2 - t-1 + 1 - t + t2 - t3 + t4 - t5 + t6 - t7 + t8 - t9 + t10 - t11 + t12 - t13 + t14 - t15 + t16

Conway Polynomial: 1 + 136z2 + 3060z4 + 27132z6 + 125970z8 + 352716z10 + 646646z12 + 817190z14 + 735471z16 + 480700z18 + 230230z20 + 80730z22 + 20475z24 + 3654z26 + 435z28 + 31z30 + z32

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

Determinant and Signature: {33, 32}

Jones Polynomial: q16 + q18 - q19 + q20 - q21 + q22 - q23 + q24 - q25 + q26 - q27 + q28 - q29 + q30 - q31 + q32 - q33 + q34 - q35 + q36 - q37 + q38 - q39 + q40 - q41 + q42 - q43 + q44 - q45 + q46 - q47 + q48 - q49

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: {136, 1496}

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=32 is the signature of T(33,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
0123456789101112131415161718192021222324252627282930313233χ
99                                 1-1
97                                  0
95                               11 0
93                                  0
91                             11   0
89                                  0
87                           11     0
85                                  0
83                         11       0
81                                  0
79                       11         0
77                                  0
75                     11           0
73                                  0
71                   11             0
69                                  0
67                 11               0
65                                  0
63               11                 0
61                                  0
59             11                   0
57                                  0
55           11                     0
53                                  0
51         11                       0
49                                  0
47       11                         0
45                                  0
43     11                           0
41                                  0
39   11                             0
37                                  0
35  1                               1
331                                 1
311                                 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[33, 2]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[33, 2]]
Out[3]=   
33
In[4]:=
PD[TorusKnot[33, 2]]
Out[4]=   
PD[X[31, 65, 32, 64], X[65, 33, 66, 32], X[33, 1, 34, 66], X[1, 35, 2, 34], 
 
>   X[35, 3, 36, 2], X[3, 37, 4, 36], X[37, 5, 38, 4], X[5, 39, 6, 38], 
 
>   X[39, 7, 40, 6], X[7, 41, 8, 40], X[41, 9, 42, 8], X[9, 43, 10, 42], 
 
>   X[43, 11, 44, 10], X[11, 45, 12, 44], X[45, 13, 46, 12], X[13, 47, 14, 46], 
 
>   X[47, 15, 48, 14], X[15, 49, 16, 48], X[49, 17, 50, 16], X[17, 51, 18, 50], 
 
>   X[51, 19, 52, 18], X[19, 53, 20, 52], X[53, 21, 54, 20], X[21, 55, 22, 54], 
 
>   X[55, 23, 56, 22], X[23, 57, 24, 56], X[57, 25, 58, 24], X[25, 59, 26, 58], 
 
>   X[59, 27, 60, 26], X[27, 61, 28, 60], X[61, 29, 62, 28], X[29, 63, 30, 62], 
 
>   X[63, 31, 64, 30]]
In[5]:=
GaussCode[TorusKnot[33, 2]]
Out[5]=   
GaussCode[-4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, 
 
>   -20, 21, -22, 23, -24, 25, -26, 27, -28, 29, -30, 31, -32, 33, -1, 2, -3, 
 
>   4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 
 
>   22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 32, -33, 1, -2, 3]
In[6]:=
BR[TorusKnot[33, 2]]
Out[6]=   
BR[2, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
 
>    1, 1, 1, 1, 1, 1, 1, 1, 1}]
In[7]:=
alex = Alexander[TorusKnot[33, 2]][t]
Out[7]=   
     -16    -15    -14    -13    -12    -11    -10    -9    -8    -7    -6
1 + t    - t    + t    - t    + t    - t    + t    - t   + t   - t   + t   - 
 
     -5    -4    -3    -2   1        2    3    4    5    6    7    8    9
>   t   + t   - t   + t   - - - t + t  - t  + t  - t  + t  - t  + t  - t  + 
                            t
 
     10    11    12    13    14    15    16
>   t   - t   + t   - t   + t   - t   + t
In[8]:=
Conway[TorusKnot[33, 2]][z]
Out[8]=   
         2         4          6           8           10           12
1 + 136 z  + 3060 z  + 27132 z  + 125970 z  + 352716 z   + 646646 z   + 
 
            14           16           18           20          22          24
>   817190 z   + 735471 z   + 480700 z   + 230230 z   + 80730 z   + 20475 z   + 
 
          26        28       30    32
>   3654 z   + 435 z   + 31 z   + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{}
In[10]:=
{KnotDet[TorusKnot[33, 2]], KnotSignature[TorusKnot[33, 2]]}
Out[10]=   
{33, 32}
In[11]:=
J=Jones[TorusKnot[33, 2]][q]
Out[11]=   
 16    18    19    20    21    22    23    24    25    26    27    28    29
q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + 
 
     30    31    32    33    34    35    36    37    38    39    40    41
>   q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + 
 
     42    43    44    45    46    47    48    49
>   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[33, 2]][q]
Out[13]=   
NotAvailable
In[14]:=
Kauffman[TorusKnot[33, 2]][a, z]
Out[14]=   
NotAvailable
In[15]:=
{Vassiliev[2][TorusKnot[33, 2]], Vassiliev[3][TorusKnot[33, 2]]}
Out[15]=   
{136, 1496}
In[16]:=
Kh[TorusKnot[33, 2]][q, t]
Out[16]=   
 31    33    35  2    39  3    39  4    43  5    43  6    47  7    47  8
q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 
 
     51  9    51  10    55  11    55  12    59  13    59  14    63  15
>   q   t  + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     63  16    67  17    67  18    71  19    71  20    75  21    75  22
>   q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     79  23    79  24    83  25    83  26    87  27    87  28    91  29
>   q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     91  30    95  31    95  32    99  33
>   q   t   + q   t   + q   t   + q   t


Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(33,2)
T(11,4)
T(11,4)
T(17,3)
T(17,3)