© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(35,2)
T(35,2)
T(3,2)
T(3,2)
T(9,5)
TubePlot
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   The 36-Crossing Torus Knot T(9,5)

Visit T(9,5)'s page at Knotilus!

Acknowledgement

PD Presentation: X51,37,52,36 X66,38,67,37 X9,39,10,38 X24,40,25,39 X67,53,68,52 X10,54,11,53 X25,55,26,54 X40,56,41,55 X11,69,12,68 X26,70,27,69 X41,71,42,70 X56,72,57,71 X27,13,28,12 X42,14,43,13 X57,15,58,14 X72,16,1,15 X43,29,44,28 X58,30,59,29 X1,31,2,30 X16,32,17,31 X59,45,60,44 X2,46,3,45 X17,47,18,46 X32,48,33,47 X3,61,4,60 X18,62,19,61 X33,63,34,62 X48,64,49,63 X19,5,20,4 X34,6,35,5 X49,7,50,6 X64,8,65,7 X35,21,36,20 X50,22,51,21 X65,23,66,22 X8,24,9,23

Gauss Code: {-19, -22, -25, 29, 30, 31, 32, -36, -3, -6, -9, 13, 14, 15, 16, -20, -23, -26, -29, 33, 34, 35, 36, -4, -7, -10, -13, 17, 18, 19, 20, -24, -27, -30, -33, 1, 2, 3, 4, -8, -11, -14, -17, 21, 22, 23, 24, -28, -31, -34, -1, 5, 6, 7, 8, -12, -15, -18, -21, 25, 26, 27, 28, -32, -35, -2, -5, 9, 10, 11, 12, -16}

Braid Representative:    

Alexander Polynomial: t-16 - t-15 + t-11 - t-10 + t-7 - t-5 + t-2 - 1 + t2 - t5 + t7 - t10 + t11 - t15 + t16

Conway Polynomial: 1 + 80z2 + 1772z4 + 17094z6 + 87560z8 + 267421z10 + 526423z12 + 703851z14 + 661810z16 + 447240z18 + 219625z20 + 78431z22 + 20150z24 + 3627z26 + 434z28 + 31z30 + z32

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

Determinant and Signature: {1, 24}

Jones Polynomial: q16 + q18 + q20 - q26 - q28

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: {80, 600}

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=24 is the signature of T(9,5). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
0123456789101112131415161718192021χ
63                    110
61                  11  0
59                1  21 0
57                131   -1
55              13  1   -1
53            12 22     -1
51             32       -1
49           32 1       0
47         2  2         0
45       1 12           0
43     1 12             0
41     11 1             1
39   11 1               1
37    1                 1
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[9, 5]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[9, 5]]
Out[3]=   
36
In[4]:=
PD[TorusKnot[9, 5]]
Out[4]=   
PD[X[51, 37, 52, 36], X[66, 38, 67, 37], X[9, 39, 10, 38], X[24, 40, 25, 39], 
 
>   X[67, 53, 68, 52], X[10, 54, 11, 53], X[25, 55, 26, 54], X[40, 56, 41, 55], 
 
>   X[11, 69, 12, 68], X[26, 70, 27, 69], X[41, 71, 42, 70], X[56, 72, 57, 71], 
 
>   X[27, 13, 28, 12], X[42, 14, 43, 13], X[57, 15, 58, 14], X[72, 16, 1, 15], 
 
>   X[43, 29, 44, 28], X[58, 30, 59, 29], X[1, 31, 2, 30], X[16, 32, 17, 31], 
 
>   X[59, 45, 60, 44], X[2, 46, 3, 45], X[17, 47, 18, 46], X[32, 48, 33, 47], 
 
>   X[3, 61, 4, 60], X[18, 62, 19, 61], X[33, 63, 34, 62], X[48, 64, 49, 63], 
 
>   X[19, 5, 20, 4], X[34, 6, 35, 5], X[49, 7, 50, 6], X[64, 8, 65, 7], 
 
>   X[35, 21, 36, 20], X[50, 22, 51, 21], X[65, 23, 66, 22], X[8, 24, 9, 23]]
In[5]:=
GaussCode[TorusKnot[9, 5]]
Out[5]=   
GaussCode[-19, -22, -25, 29, 30, 31, 32, -36, -3, -6, -9, 13, 14, 15, 16, -20, 
 
>   -23, -26, -29, 33, 34, 35, 36, -4, -7, -10, -13, 17, 18, 19, 20, -24, -27, 
 
>   -30, -33, 1, 2, 3, 4, -8, -11, -14, -17, 21, 22, 23, 24, -28, -31, -34, -1, 
 
>   5, 6, 7, 8, -12, -15, -18, -21, 25, 26, 27, 28, -32, -35, -2, -5, 9, 10, 
 
>   11, 12, -16]
In[6]:=
BR[TorusKnot[9, 5]]
Out[6]=   
BR[5, {1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 
 
>    1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4}]
In[7]:=
alex = Alexander[TorusKnot[9, 5]][t]
Out[7]=   
      -16    -15    -11    -10    -7    -5    -2    2    5    7    10    11
-1 + t    - t    + t    - t    + t   - t   + t   + t  - t  + t  - t   + t   - 
 
     15    16
>   t   + t
In[8]:=
Conway[TorusKnot[9, 5]][z]
Out[8]=   
        2         4          6          8           10           12
1 + 80 z  + 1772 z  + 17094 z  + 87560 z  + 267421 z   + 526423 z   + 
 
            14           16           18           20          22          24
>   703851 z   + 661810 z   + 447240 z   + 219625 z   + 78431 z   + 20150 z   + 
 
          26        28       30    32
>   3627 z   + 434 z   + 31 z   + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{}
In[10]:=
{KnotDet[TorusKnot[9, 5]], KnotSignature[TorusKnot[9, 5]]}
Out[10]=   
{1, 24}
In[11]:=
J=Jones[TorusKnot[9, 5]][q]
Out[11]=   
 16    18    20    26    28
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, 5]][q]
Out[13]=   
NotAvailable
In[14]:=
Kauffman[TorusKnot[9, 5]][a, z]
Out[14]=   
NotAvailable
In[15]:=
{Vassiliev[2][TorusKnot[9, 5]], Vassiliev[3][TorusKnot[9, 5]]}
Out[15]=   
{80, 600}
In[16]:=
Kh[TorusKnot[9, 5]][q, t]
Out[16]=   
 31    33    35  2    39  3    37  4    39  4    41  5    43  5    39  6
q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 
 
     41  6    43  7    45  7    41  8      43  8    45  9      47  9
>   q   t  + q   t  + q   t  + q   t  + 2 q   t  + q   t  + 2 q   t  + 
 
       45  10      49  11      47  12      49  12    53  12      51  13
>   2 q   t   + 3 q   t   + 2 q   t   + 2 q   t   + q   t   + 3 q   t   + 
 
       53  13    49  14      51  14    55  14      53  15      55  15
>   2 q   t   + q   t   + 2 q   t   + q   t   + 2 q   t   + 3 q   t   + 
 
       53  16    57  16    59  16      57  17    55  18    57  18    61  18
>   2 q   t   + q   t   + q   t   + 3 q   t   + q   t   + q   t   + q   t   + 
 
       59  19    61  19    59  20    63  20    63  21
>   2 q   t   + q   t   + q   t   + q   t   + q   t


Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(9,5)
T(35,2)
T(35,2)
T(3,2)
T(3,2)