© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(33,2)
T(33,2)
T(7,6)
T(7,6)
T(17,3)
TubePlot
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   The 34-Crossing Torus Knot T(17,3)

Visit T(17,3)'s page at Knotilus!

Acknowledgement

PD Presentation: X66,44,67,43 X21,45,22,44 X22,68,23,67 X45,1,46,68 X46,24,47,23 X1,25,2,24 X2,48,3,47 X25,49,26,48 X26,4,27,3 X49,5,50,4 X50,28,51,27 X5,29,6,28 X6,52,7,51 X29,53,30,52 X30,8,31,7 X53,9,54,8 X54,32,55,31 X9,33,10,32 X10,56,11,55 X33,57,34,56 X34,12,35,11 X57,13,58,12 X58,36,59,35 X13,37,14,36 X14,60,15,59 X37,61,38,60 X38,16,39,15 X61,17,62,16 X62,40,63,39 X17,41,18,40 X18,64,19,63 X41,65,42,64 X42,20,43,19 X65,21,66,20

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

Braid Representative:    

Alexander Polynomial: t-16 - t-15 + t-13 - t-12 + t-10 - t-9 + t-7 - t-6 + t-4 - t-3 + t-1 - 1 + t - t3 + t4 - t6 + t7 - t9 + t10 - t12 + t13 - t15 + t16

Conway Polynomial: 1 + 96z2 + 2280z4 + 21204z6 + 102752z8 + 298870z10 + 566618z12 + 737276z14 + 680390z16 + 454195z18 + 221355z20 + 78705z22 + 20175z24 + 3628z26 + 434z28 + 31z30 + z32

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

Determinant and Signature: {1, 24}

Jones Polynomial: q16 + q18 - q34

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: {96, 816}

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(17,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
01234567891011121314151617181920212223χ
69                       1-1
67                     1  -1
65                     11 0
63                   11   0
61                 1  1   0
59                 11     0
57               11       0
55             1  1       0
53             11         0
51           11           0
49         1  1           0
47         11             0
45       11               0
43     1  1               0
41     11                 0
39   11                   0
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[17, 3]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[17, 3]]
Out[3]=   
34
In[4]:=
PD[TorusKnot[17, 3]]
Out[4]=   
PD[X[66, 44, 67, 43], X[21, 45, 22, 44], X[22, 68, 23, 67], X[45, 1, 46, 68], 
 
>   X[46, 24, 47, 23], X[1, 25, 2, 24], X[2, 48, 3, 47], X[25, 49, 26, 48], 
 
>   X[26, 4, 27, 3], X[49, 5, 50, 4], X[50, 28, 51, 27], X[5, 29, 6, 28], 
 
>   X[6, 52, 7, 51], X[29, 53, 30, 52], X[30, 8, 31, 7], X[53, 9, 54, 8], 
 
>   X[54, 32, 55, 31], X[9, 33, 10, 32], X[10, 56, 11, 55], X[33, 57, 34, 56], 
 
>   X[34, 12, 35, 11], X[57, 13, 58, 12], X[58, 36, 59, 35], X[13, 37, 14, 36], 
 
>   X[14, 60, 15, 59], X[37, 61, 38, 60], X[38, 16, 39, 15], X[61, 17, 62, 16], 
 
>   X[62, 40, 63, 39], X[17, 41, 18, 40], X[18, 64, 19, 63], X[41, 65, 42, 64], 
 
>   X[42, 20, 43, 19], X[65, 21, 66, 20]]
In[5]:=
GaussCode[TorusKnot[17, 3]]
Out[5]=   
GaussCode[-6, -7, 9, 10, -12, -13, 15, 16, -18, -19, 21, 22, -24, -25, 27, 28, 
 
>   -30, -31, 33, 34, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 17, 18, -20, -21, 
 
>   23, 24, -26, -27, 29, 30, -32, -33, 1, 2, -4, -5, 7, 8, -10, -11, 13, 14, 
 
>   -16, -17, 19, 20, -22, -23, 25, 26, -28, -29, 31, 32, -34, -1, 3, 4]
In[6]:=
BR[TorusKnot[17, 3]]
Out[6]=   
BR[3, {1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 
 
>    1, 2, 1, 2, 1, 2, 1, 2, 1, 2}]
In[7]:=
alex = Alexander[TorusKnot[17, 3]][t]
Out[7]=   
      -16    -15    -13    -12    -10    -9    -7    -6    -4    -3   1
-1 + t    - t    + t    - t    + t    - t   + t   - t   + t   - t   + - + t - 
                                                                      t
 
     3    4    6    7    9    10    12    13    15    16
>   t  + t  - t  + t  - t  + t   - t   + t   - t   + t
In[8]:=
Conway[TorusKnot[17, 3]][z]
Out[8]=   
        2         4          6           8           10           12
1 + 96 z  + 2280 z  + 21204 z  + 102752 z  + 298870 z   + 566618 z   + 
 
            14           16           18           20          22          24
>   737276 z   + 680390 z   + 454195 z   + 221355 z   + 78705 z   + 20175 z   + 
 
          26        28       30    32
>   3628 z   + 434 z   + 31 z   + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{}
In[10]:=
{KnotDet[TorusKnot[17, 3]], KnotSignature[TorusKnot[17, 3]]}
Out[10]=   
{1, 24}
In[11]:=
J=Jones[TorusKnot[17, 3]][q]
Out[11]=   
 16    18    34
q   + q   - q
In[12]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[12]=   
{}
In[13]:=
A2Invariant[TorusKnot[17, 3]][q]
Out[13]=   
NotAvailable
In[14]:=
Kauffman[TorusKnot[17, 3]][a, z]
Out[14]=   
NotAvailable
In[15]:=
{Vassiliev[2][TorusKnot[17, 3]], Vassiliev[3][TorusKnot[17, 3]]}
Out[15]=   
{96, 816}
In[16]:=
Kh[TorusKnot[17, 3]][q, t]
Out[16]=   
 31    33    35  2    39  3    37  4    39  4    41  5    43  5    41  6
q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 
 
     45  7    43  8    45  8    47  9    49  9    47  10    51  11    49  12
>   q   t  + q   t  + q   t  + q   t  + q   t  + q   t   + q   t   + q   t   + 
 
     51  12    53  13    55  13    53  14    57  15    55  16    57  16
>   q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     59  17    61  17    59  18    63  19    61  20    63  20    65  21
>   q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     67  21    65  22    69  23
>   q   t   + q   t   + q   t


Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(17,3)
T(33,2)
T(33,2)
T(7,6)
T(7,6)