© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(8,3)
T(8,3) 		T(19,2)
T(19,2)
T(17,2)
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   The 17-Crossing Torus Knot T(17,2)

Visit T(17,2)'s page at Knotilus!

Acknowledgement

PD Presentation: X15,33,16,32 X33,17,34,16 X17,1,18,34 X1,19,2,18 X19,3,20,2 X3,21,4,20 X21,5,22,4 X5,23,6,22 X23,7,24,6 X7,25,8,24 X25,9,26,8 X9,27,10,26 X27,11,28,10 X11,29,12,28 X29,13,30,12 X13,31,14,30 X31,15,32,14

Gauss Code: {-4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 1, -2, 3}

Braid Representative:    

Alexander Polynomial: 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

Conway Polynomial: 1 + 36z2 + 210z4 + 462z6 + 495z8 + 286z10 + 91z12 + 15z14 + z16

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

Determinant and Signature: {17, 16}

Jones Polynomial: q8 + q10 - q11 + q12 - q13 + q14 - q15 + q16 - q17 + q18 - q19 + q20 - q21 + q22 - q23 + q24 - q25

Other knots (up to mirrors) with the same Jones Polynomial: {...}

Coloured Jones Polynomial (in the 3-dimensional representation of sl(2); n=2): q16 + q19 - q21 + q22 - q24 + q25 - q27 + q28 - q30 + q31 - q33 + q34 - q36 + q37 - q39 + q40 - q42 + q43 - q45 + q46 - q48 + q49 - 2q51 + q52 - q54 + q55 - q57 + q58 - q60 + q61 - q63 + q64 - q66 + q67

A2 (sl(3)) Invariant: Not Available.

Kauffman Polynomial: Not Available.

V2 and V3, the type 2 and 3 Vassiliev invariants: {36, 204}

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=16 is the signature of T(17,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 01234567891011121314151617χ
51                 1-1
49                  0
47               11 0
45                  0
43             11   0
41                  0
39           11     0
37                  0
35         11       0
33                  0
31       11         0
29                  0
27     11           0
25                  0
23   11             0
21                  0
19  1               1
171                 1
151                 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[17, 2]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[17, 2]]
Out[3]=   
17
In[4]:=
PD[TorusKnot[17, 2]]
Out[4]=   
PD[X[15, 33, 16, 32], X[33, 17, 34, 16], X[17, 1, 18, 34], X[1, 19, 2, 18], 
 
>   X[19, 3, 20, 2], X[3, 21, 4, 20], X[21, 5, 22, 4], X[5, 23, 6, 22], 
 
>   X[23, 7, 24, 6], X[7, 25, 8, 24], X[25, 9, 26, 8], X[9, 27, 10, 26], 
 
>   X[27, 11, 28, 10], X[11, 29, 12, 28], X[29, 13, 30, 12], X[13, 31, 14, 30], 
 
>   X[31, 15, 32, 14]]
In[5]:=
GaussCode[TorusKnot[17, 2]]
Out[5]=   
GaussCode[-4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -1, 2, -3, 
 
>   4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 1, -2, 3]
In[6]:=
BR[TorusKnot[17, 2]]
Out[6]=   
BR[2, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}]
In[7]:=
alex = Alexander[TorusKnot[17, 2]][t]
Out[7]=   
     -8    -7    -6    -5    -4    -3    -2   1        2    3    4    5    6
1 + t   - t   + t   - t   + t   - t   + t   - - - t + t  - t  + t  - t  + t  - 
                                              t
 
     7    8
>   t  + t
In[8]:=
Conway[TorusKnot[17, 2]][z]
Out[8]=   
        2        4        6        8        10       12       14    16
1 + 36 z  + 210 z  + 462 z  + 495 z  + 286 z   + 91 z   + 15 z   + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{}
In[10]:=
{KnotDet[TorusKnot[17, 2]], KnotSignature[TorusKnot[17, 2]]}
Out[10]=   
{17, 16}
In[11]:=
J=Jones[TorusKnot[17, 2]][q]
Out[11]=   
 8    10    11    12    13    14    15    16    17    18    19    20    21
q  + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + 
 
     22    23    24    25
>   q   - q   + q   - q
In[12]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[12]=   
{}
In[13]:=
ColouredJones[TorusKnot[17, 2], 2][q]
Out[13]=   
 16    19    21    22    24    25    27    28    30    31    33    34    36
q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + 
 
     37    39    40    42    43    45    46    48    49      51    52    54
>   q   - q   + q   - q   + q   - q   + q   - q   + q   - 2 q   + q   - q   + 
 
     55    57    58    60    61    63    64    66    67
>   q   - q   + q   - q   + q   - q   + q   - q   + q
In[14]:=
A2Invariant[TorusKnot[17, 2]][q]
Out[14]=   
NotAvailable
In[15]:=
Kauffman[TorusKnot[17, 2]][a, z]
Out[15]=   
NotAvailable
In[16]:=
{Vassiliev[2][TorusKnot[17, 2]], Vassiliev[3][TorusKnot[17, 2]]}
Out[16]=   
{36, 204}
In[17]:=
Kh[TorusKnot[17, 2]][q, t]
Out[17]=   
 15    17    19  2    23  3    23  4    27  5    27  6    31  7    31  8
q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 
 
     35  9    35  10    39  11    39  12    43  13    43  14    47  15
>   q   t  + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     47  16    51  17
>   q   t   + q   t

Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(17,2)
T(8,3)
T(8,3) 		T(19,2)
T(19,2)