© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(31,2)
T(31,2)
T(16,3)
T(16,3)
T(8,5)
TubePlot
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   The 32-Crossing Torus Knot T(8,5)

Visit T(8,5)'s page at Knotilus!

Acknowledgement

PD Presentation: X54,16,55,15 X29,17,30,16 X4,18,5,17 X43,19,44,18 X30,56,31,55 X5,57,6,56 X44,58,45,57 X19,59,20,58 X6,32,7,31 X45,33,46,32 X20,34,21,33 X59,35,60,34 X46,8,47,7 X21,9,22,8 X60,10,61,9 X35,11,36,10 X22,48,23,47 X61,49,62,48 X36,50,37,49 X11,51,12,50 X62,24,63,23 X37,25,38,24 X12,26,13,25 X51,27,52,26 X38,64,39,63 X13,1,14,64 X52,2,53,1 X27,3,28,2 X14,40,15,39 X53,41,54,40 X28,42,29,41 X3,43,4,42

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

Braid Representative:    

Alexander Polynomial: t-14 - t-13 + t-9 - t-8 + t-6 - t-5 + t-4 - t-3 + t-1 - 1 + t - t3 + t4 - t5 + t6 - t8 + t9 - t13 + t14

Conway Polynomial: 1 + 63z2 + 1092z4 + 8169z6 + 32055z8 + 73876z10 + 107849z12 + 104771z14 + 69785z16 + 32320z18 + 10395z20 + 2277z22 + 324z24 + 27z26 + z28

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

Determinant and Signature: {5, 20}

Jones Polynomial: q14 + q16 + q18 - q23 - q25

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: {63, 420}

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=20 is the signature of T(8,5). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
012345678910111213141516171819χ
57                  110
55                11  0
53                121 0
51              131   -1
49            12 11   -1
47             32     -1
45           32       -1
43         2  2       0
41       1 12         0
39     1 12           0
37     11 1           1
35   11 1             1
33    1               1
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[8, 5]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[8, 5]]
Out[3]=   
32
In[4]:=
PD[TorusKnot[8, 5]]
Out[4]=   
PD[X[54, 16, 55, 15], X[29, 17, 30, 16], X[4, 18, 5, 17], X[43, 19, 44, 18], 
 
>   X[30, 56, 31, 55], X[5, 57, 6, 56], X[44, 58, 45, 57], X[19, 59, 20, 58], 
 
>   X[6, 32, 7, 31], X[45, 33, 46, 32], X[20, 34, 21, 33], X[59, 35, 60, 34], 
 
>   X[46, 8, 47, 7], X[21, 9, 22, 8], X[60, 10, 61, 9], X[35, 11, 36, 10], 
 
>   X[22, 48, 23, 47], X[61, 49, 62, 48], X[36, 50, 37, 49], X[11, 51, 12, 50], 
 
>   X[62, 24, 63, 23], X[37, 25, 38, 24], X[12, 26, 13, 25], X[51, 27, 52, 26], 
 
>   X[38, 64, 39, 63], X[13, 1, 14, 64], X[52, 2, 53, 1], X[27, 3, 28, 2], 
 
>   X[14, 40, 15, 39], X[53, 41, 54, 40], X[28, 42, 29, 41], X[3, 43, 4, 42]]
In[5]:=
GaussCode[TorusKnot[8, 5]]
Out[5]=   
GaussCode[27, 28, -32, -3, -6, -9, 13, 14, 15, 16, -20, -23, -26, -29, 1, 2, 3, 
 
>   4, -8, -11, -14, -17, 21, 22, 23, 24, -28, -31, -2, -5, 9, 10, 11, 12, -16, 
 
>   -19, -22, -25, 29, 30, 31, 32, -4, -7, -10, -13, 17, 18, 19, 20, -24, -27, 
 
>   -30, -1, 5, 6, 7, 8, -12, -15, -18, -21, 25, 26]
In[6]:=
BR[TorusKnot[8, 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}]
In[7]:=
alex = Alexander[TorusKnot[8, 5]][t]
Out[7]=   
      -14    -13    -9    -8    -6    -5    -4    -3   1        3    4    5
-1 + t    - t    + t   - t   + t   - t   + t   - t   + - + t - t  + t  - t  + 
                                                       t
 
     6    8    9    13    14
>   t  - t  + t  - t   + t
In[8]:=
Conway[TorusKnot[8, 5]][z]
Out[8]=   
        2         4         6          8          10           12
1 + 63 z  + 1092 z  + 8169 z  + 32055 z  + 73876 z   + 107849 z   + 
 
            14          16          18          20         22        24
>   104771 z   + 69785 z   + 32320 z   + 10395 z   + 2277 z   + 324 z   + 
 
        26    28
>   27 z   + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{}
In[10]:=
{KnotDet[TorusKnot[8, 5]], KnotSignature[TorusKnot[8, 5]]}
Out[10]=   
{5, 20}
In[11]:=
J=Jones[TorusKnot[8, 5]][q]
Out[11]=   
 14    16    18    23    25
q   + q   + q   - q   - q
In[12]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[12]=   
{}
In[13]:=
A2Invariant[TorusKnot[8, 5]][q]
Out[13]=   
NotAvailable
In[14]:=
Kauffman[TorusKnot[8, 5]][a, z]
Out[14]=   
NotAvailable
In[15]:=
{Vassiliev[2][TorusKnot[8, 5]], Vassiliev[3][TorusKnot[8, 5]]}
Out[15]=   
{63, 420}
In[16]:=
Kh[TorusKnot[8, 5]][q, t]
Out[16]=   
 27    29    31  2    35  3    33  4    35  4    37  5    39  5    35  6
q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 
 
     37  6    39  7    41  7    37  8      39  8    41  9      43  9
>   q   t  + q   t  + q   t  + q   t  + 2 q   t  + q   t  + 2 q   t  + 
 
       41  10      45  11      43  12      45  12    49  12      47  13
>   2 q   t   + 3 q   t   + 2 q   t   + 2 q   t   + q   t   + 3 q   t   + 
 
       49  13      47  14    51  14    49  15      51  15    49  16    51  16
>   2 q   t   + 2 q   t   + q   t   + q   t   + 3 q   t   + q   t   + q   t   + 
 
     53  16    55  16      53  17    55  17    53  18    57  18    57  19
>   q   t   + q   t   + 2 q   t   + q   t   + q   t   + q   t   + q   t


Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(8,5)
T(31,2)
T(31,2)
T(16,3)
T(16,3)