© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(7,6)
T(7,6)
T(9,5)
T(9,5)
T(35,2)
TubePlot
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   The 35-Crossing Torus Knot T(35,2)

Visit T(35,2)'s page at Knotilus!

Acknowledgement

PD Presentation: X29,65,30,64 X65,31,66,30 X31,67,32,66 X67,33,68,32 X33,69,34,68 X69,35,70,34 X35,1,36,70 X1,37,2,36 X37,3,38,2 X3,39,4,38 X39,5,40,4 X5,41,6,40 X41,7,42,6 X7,43,8,42 X43,9,44,8 X9,45,10,44 X45,11,46,10 X11,47,12,46 X47,13,48,12 X13,49,14,48 X49,15,50,14 X15,51,16,50 X51,17,52,16 X17,53,18,52 X53,19,54,18 X19,55,20,54 X55,21,56,20 X21,57,22,56 X57,23,58,22 X23,59,24,58 X59,25,60,24 X25,61,26,60 X61,27,62,26 X27,63,28,62 X63,29,64,28

Gauss Code: {-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, -34, 35, -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, 34, -35, 1, -2, 3, -4, 5, -6, 7}

Braid Representative:    

Alexander Polynomial: t-17 - 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 + t17

Conway Polynomial: 1 + 153z2 + 3876z4 + 38760z6 + 203490z8 + 646646z10 + 1352078z12 + 1961256z14 + 2042975z16 + 1562275z18 + 888030z20 + 376740z22 + 118755z24 + 27405z26 + 4495z28 + 496z30 + 33z32 + z34

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

Determinant and Signature: {35, 34}

Jones Polynomial: q17 + 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 - q50 + q51 - q52

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: {153, 1785}

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=34 is the signature of T(35,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
01234567891011121314151617181920212223242526272829303132333435χ
105                                   1-1
103                                    0
101                                 11 0
99                                    0
97                               11   0
95                                    0
93                             11     0
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  1                                 1
351                                   1
331                                   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[35, 2]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[35, 2]]
Out[3]=   
35
In[4]:=
PD[TorusKnot[35, 2]]
Out[4]=   
PD[X[29, 65, 30, 64], X[65, 31, 66, 30], X[31, 67, 32, 66], X[67, 33, 68, 32], 
 
>   X[33, 69, 34, 68], X[69, 35, 70, 34], X[35, 1, 36, 70], X[1, 37, 2, 36], 
 
>   X[37, 3, 38, 2], X[3, 39, 4, 38], X[39, 5, 40, 4], X[5, 41, 6, 40], 
 
>   X[41, 7, 42, 6], X[7, 43, 8, 42], X[43, 9, 44, 8], X[9, 45, 10, 44], 
 
>   X[45, 11, 46, 10], X[11, 47, 12, 46], X[47, 13, 48, 12], X[13, 49, 14, 48], 
 
>   X[49, 15, 50, 14], X[15, 51, 16, 50], X[51, 17, 52, 16], X[17, 53, 18, 52], 
 
>   X[53, 19, 54, 18], X[19, 55, 20, 54], X[55, 21, 56, 20], X[21, 57, 22, 56], 
 
>   X[57, 23, 58, 22], X[23, 59, 24, 58], X[59, 25, 60, 24], X[25, 61, 26, 60], 
 
>   X[61, 27, 62, 26], X[27, 63, 28, 62], X[63, 29, 64, 28]]
In[5]:=
GaussCode[TorusKnot[35, 2]]
Out[5]=   
GaussCode[-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, -34, 35, -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, 34, -35, 1, -2, 3, -4, 5, -6, 
 
>   7]
In[6]:=
BR[TorusKnot[35, 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, 1, 1}]
In[7]:=
alex = Alexander[TorusKnot[35, 2]][t]
Out[7]=   
      -17    -16    -15    -14    -13    -12    -11    -10    -9    -8    -7
-1 + t    - t    + t    - t    + t    - t    + t    - t    + t   - t   + t   - 
 
     -6    -5    -4    -3    -2   1        2    3    4    5    6    7    8
>   t   + t   - t   + t   - t   + - + t - t  + t  - t  + t  - t  + t  - t  + 
                                  t
 
     9    10    11    12    13    14    15    16    17
>   t  - t   + t   - t   + t   - t   + t   - t   + t
In[8]:=
Conway[TorusKnot[35, 2]][z]
Out[8]=   
         2         4          6           8           10            12
1 + 153 z  + 3876 z  + 38760 z  + 203490 z  + 646646 z   + 1352078 z   + 
 
             14            16            18           20           22
>   1961256 z   + 2042975 z   + 1562275 z   + 888030 z   + 376740 z   + 
 
            24          26         28        30       32    34
>   118755 z   + 27405 z   + 4495 z   + 496 z   + 33 z   + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{}
In[10]:=
{KnotDet[TorusKnot[35, 2]], KnotSignature[TorusKnot[35, 2]]}
Out[10]=   
{35, 34}
In[11]:=
J=Jones[TorusKnot[35, 2]][q]
Out[11]=   
 17    19    20    21    22    23    24    25    26    27    28    29    30
q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + 
 
     31    32    33    34    35    36    37    38    39    40    41    42
>   q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + 
 
     43    44    45    46    47    48    49    50    51    52
>   q   - 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[35, 2]][q]
Out[13]=   
NotAvailable
In[14]:=
Kauffman[TorusKnot[35, 2]][a, z]
Out[14]=   
NotAvailable
In[15]:=
{Vassiliev[2][TorusKnot[35, 2]], Vassiliev[3][TorusKnot[35, 2]]}
Out[15]=   
{153, 1785}
In[16]:=
Kh[TorusKnot[35, 2]][q, t]
Out[16]=   
 33    35    37  2    41  3    41  4    45  5    45  6    49  7    49  8
q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 
 
     53  9    53  10    57  11    57  12    61  13    61  14    65  15
>   q   t  + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     65  16    69  17    69  18    73  19    73  20    77  21    77  22
>   q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     81  23    81  24    85  25    85  26    89  27    89  28    93  29
>   q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     93  30    97  31    97  32    101  33    101  34    105  35
>   q   t   + q   t   + q   t   + q    t   + q    t   + q    t


Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(35,2)
T(7,6)
T(7,6)
T(9,5)
T(9,5)