© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(29,2)
T(29,2)
T(8,5)
T(8,5)
T(31,2)
TubePlot
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   The 31-Crossing Torus Knot T(31,2)

Visit T(31,2)'s page at Knotilus!

Acknowledgement

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

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

Braid Representative:    

Alexander Polynomial: 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

Conway Polynomial: 1 + 120z2 + 2380z4 + 18564z6 + 75582z8 + 184756z10 + 293930z12 + 319770z14 + 245157z16 + 134596z18 + 53130z20 + 14950z22 + 2925z24 + 378z26 + 29z28 + z30

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

Determinant and Signature: {31, 30}

Jones Polynomial: q15 + 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 - q44 + q45 - q46

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: {120, 1240}

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=30 is the signature of T(31,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
012345678910111213141516171819202122232425262728293031χ
93                               1-1
91                                0
89                             11 0
87                                0
85                           11   0
83                                0
81                         11     0
79                                0
77                       11       0
75                                0
73                     11         0
71                                0
69                   11           0
67                                0
65                 11             0
63                                0
61               11               0
59                                0
57             11                 0
55                                0
53           11                   0
51                                0
49         11                     0
47                                0
45       11                       0
43                                0
41     11                         0
39                                0
37   11                           0
35                                0
33  1                             1
311                               1
291                               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[31, 2]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[31, 2]]
Out[3]=   
31
In[4]:=
PD[TorusKnot[31, 2]]
Out[4]=   
PD[X[17, 49, 18, 48], X[49, 19, 50, 18], X[19, 51, 20, 50], X[51, 21, 52, 20], 
 
>   X[21, 53, 22, 52], X[53, 23, 54, 22], X[23, 55, 24, 54], X[55, 25, 56, 24], 
 
>   X[25, 57, 26, 56], X[57, 27, 58, 26], X[27, 59, 28, 58], X[59, 29, 60, 28], 
 
>   X[29, 61, 30, 60], X[61, 31, 62, 30], X[31, 1, 32, 62], X[1, 33, 2, 32], 
 
>   X[33, 3, 34, 2], X[3, 35, 4, 34], X[35, 5, 36, 4], X[5, 37, 6, 36], 
 
>   X[37, 7, 38, 6], X[7, 39, 8, 38], X[39, 9, 40, 8], X[9, 41, 10, 40], 
 
>   X[41, 11, 42, 10], X[11, 43, 12, 42], X[43, 13, 44, 12], X[13, 45, 14, 44], 
 
>   X[45, 15, 46, 14], X[15, 47, 16, 46], X[47, 17, 48, 16]]
In[5]:=
GaussCode[TorusKnot[31, 2]]
Out[5]=   
GaussCode[-16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -28, 29, -30, 
 
>   31, -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, 1, -2, 3, -4, 5, 
 
>   -6, 7, -8, 9, -10, 11, -12, 13, -14, 15]
In[6]:=
BR[TorusKnot[31, 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}]
In[7]:=
alex = Alexander[TorusKnot[31, 2]][t]
Out[7]=   
      -15    -14    -13    -12    -11    -10    -9    -8    -7    -6    -5
-1 + t    - t    + t    - t    + t    - t    + t   - t   + t   - t   + t   - 
 
     -4    -3    -2   1        2    3    4    5    6    7    8    9    10
>   t   + t   - t   + - + t - t  + t  - t  + t  - t  + t  - t  + t  - t   + 
                      t
 
     11    12    13    14    15
>   t   - t   + t   - t   + t
In[8]:=
Conway[TorusKnot[31, 2]][z]
Out[8]=   
         2         4          6          8           10           12
1 + 120 z  + 2380 z  + 18564 z  + 75582 z  + 184756 z   + 293930 z   + 
 
            14           16           18          20          22         24
>   319770 z   + 245157 z   + 134596 z   + 53130 z   + 14950 z   + 2925 z   + 
 
         26       28    30
>   378 z   + 29 z   + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{}
In[10]:=
{KnotDet[TorusKnot[31, 2]], KnotSignature[TorusKnot[31, 2]]}
Out[10]=   
{31, 30}
In[11]:=
J=Jones[TorusKnot[31, 2]][q]
Out[11]=   
 15    17    18    19    20    21    22    23    24    25    26    27    28
q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + 
 
     29    30    31    32    33    34    35    36    37    38    39    40
>   q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + 
 
     41    42    43    44    45    46
>   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[31, 2]][q]
Out[13]=   
NotAvailable
In[14]:=
Kauffman[TorusKnot[31, 2]][a, z]
Out[14]=   
NotAvailable
In[15]:=
{Vassiliev[2][TorusKnot[31, 2]], Vassiliev[3][TorusKnot[31, 2]]}
Out[15]=   
{120, 1240}
In[16]:=
Kh[TorusKnot[31, 2]][q, t]
Out[16]=   
 29    31    33  2    37  3    37  4    41  5    41  6    45  7    45  8
q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 
 
     49  9    49  10    53  11    53  12    57  13    57  14    61  15
>   q   t  + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     61  16    65  17    65  18    69  19    69  20    73  21    73  22
>   q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     77  23    77  24    81  25    81  26    85  27    85  28    89  29
>   q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     89  30    93  31
>   q   t   + q   t


Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(31,2)
T(29,2)
T(29,2)
T(8,5)
T(8,5)