© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(14,3)
T(14,3)
T(31,2)
T(31,2)
T(29,2)
TubePlot
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   The 29-Crossing Torus Knot T(29,2)

Visit T(29,2)'s page at Knotilus!

Acknowledgement

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

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

Braid Representative:    

Alexander Polynomial: 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

Conway Polynomial: 1 + 105z2 + 1820z4 + 12376z6 + 43758z8 + 92378z10 + 125970z12 + 116280z14 + 74613z16 + 33649z18 + 10626z20 + 2300z22 + 325z24 + 27z26 + z28

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

Determinant and Signature: {29, 28}

Jones Polynomial: q14 + q16 - q17 + 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

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: {105, 1015}

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=28 is the signature of T(29,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
01234567891011121314151617181920212223242526272829χ
87                             1-1
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   11                         0
33                              0
31  1                           1
291                             1
271                             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[29, 2]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[29, 2]]
Out[3]=   
29
In[4]:=
PD[TorusKnot[29, 2]]
Out[4]=   
PD[X[19, 49, 20, 48], X[49, 21, 50, 20], X[21, 51, 22, 50], X[51, 23, 52, 22], 
 
>   X[23, 53, 24, 52], X[53, 25, 54, 24], X[25, 55, 26, 54], X[55, 27, 56, 26], 
 
>   X[27, 57, 28, 56], X[57, 29, 58, 28], X[29, 1, 30, 58], X[1, 31, 2, 30], 
 
>   X[31, 3, 32, 2], X[3, 33, 4, 32], X[33, 5, 34, 4], X[5, 35, 6, 34], 
 
>   X[35, 7, 36, 6], X[7, 37, 8, 36], X[37, 9, 38, 8], X[9, 39, 10, 38], 
 
>   X[39, 11, 40, 10], X[11, 41, 12, 40], X[41, 13, 42, 12], X[13, 43, 14, 42], 
 
>   X[43, 15, 44, 14], X[15, 45, 16, 44], X[45, 17, 46, 16], X[17, 47, 18, 46], 
 
>   X[47, 19, 48, 18]]
In[5]:=
GaussCode[TorusKnot[29, 2]]
Out[5]=   
GaussCode[-12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 
 
>   27, -28, 29, -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, 1, -2, 3, -4, 5, 
 
>   -6, 7, -8, 9, -10, 11]
In[6]:=
BR[TorusKnot[29, 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}]
In[7]:=
alex = Alexander[TorusKnot[29, 2]][t]
Out[7]=   
     -14    -13    -12    -11    -10    -9    -8    -7    -6    -5    -4
1 + t    - t    + t    - t    + t    - t   + t   - t   + t   - t   + t   - 
 
     -3    -2   1        2    3    4    5    6    7    8    9    10    11
>   t   + t   - - - t + t  - t  + t  - t  + t  - t  + t  - t  + t   - t   + 
                t
 
     12    13    14
>   t   - t   + t
In[8]:=
Conway[TorusKnot[29, 2]][z]
Out[8]=   
         2         4          6          8          10           12
1 + 105 z  + 1820 z  + 12376 z  + 43758 z  + 92378 z   + 125970 z   + 
 
            14          16          18          20         22        24
>   116280 z   + 74613 z   + 33649 z   + 10626 z   + 2300 z   + 325 z   + 
 
        26    28
>   27 z   + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{}
In[10]:=
{KnotDet[TorusKnot[29, 2]], KnotSignature[TorusKnot[29, 2]]}
Out[10]=   
{29, 28}
In[11]:=
J=Jones[TorusKnot[29, 2]][q]
Out[11]=   
 14    16    17    18    19    20    21    22    23    24    25    26    27
q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + 
 
     28    29    30    31    32    33    34    35    36    37    38    39
>   q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + 
 
     40    41    42    43
>   q   - q   + q   - q
In[12]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[12]=   
{}
In[13]:=
A2Invariant[TorusKnot[29, 2]][q]
Out[13]=   
NotAvailable
In[14]:=
Kauffman[TorusKnot[29, 2]][a, z]
Out[14]=   
NotAvailable
In[15]:=
{Vassiliev[2][TorusKnot[29, 2]], Vassiliev[3][TorusKnot[29, 2]]}
Out[15]=   
{105, 1015}
In[16]:=
Kh[TorusKnot[29, 2]][q, t]
Out[16]=   
 27    29    31  2    35  3    35  4    39  5    39  6    43  7    43  8
q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 
 
     47  9    47  10    51  11    51  12    55  13    55  14    59  15
>   q   t  + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     59  16    63  17    63  18    67  19    67  20    71  21    71  22
>   q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     75  23    75  24    79  25    79  26    83  27    83  28    87  29
>   q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t


Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(29,2)
T(14,3)
T(14,3)
T(31,2)
T(31,2)