© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(8,5)
T(8,5)
T(11,4)
T(11,4)
T(16,3)
TubePlot
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   The 32-Crossing Torus Knot T(16,3)

Visit T(16,3)'s page at Knotilus!

Acknowledgement

PD Presentation: X41,63,42,62 X20,64,21,63 X21,43,22,42 X64,44,1,43 X1,23,2,22 X44,24,45,23 X45,3,46,2 X24,4,25,3 X25,47,26,46 X4,48,5,47 X5,27,6,26 X48,28,49,27 X49,7,50,6 X28,8,29,7 X29,51,30,50 X8,52,9,51 X9,31,10,30 X52,32,53,31 X53,11,54,10 X32,12,33,11 X33,55,34,54 X12,56,13,55 X13,35,14,34 X56,36,57,35 X57,15,58,14 X36,16,37,15 X37,59,38,58 X16,60,17,59 X17,39,18,38 X60,40,61,39 X61,19,62,18 X40,20,41,19

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

Braid Representative:    

Alexander Polynomial: t-15 - t-14 + t-12 - t-11 + t-9 - t-8 + t-6 - t-5 + t-3 - t-2 + 1 - t2 + t3 - t5 + t6 - t8 + t9 - t11 + t12 - t14 + t15

Conway Polynomial: 1 + 85z2 + 1785z4 + 14637z6 + 62305z8 + 158389z10 + 260729z12 + 292077z14 + 229517z16 + 128593z18 + 51589z20 + 14697z22 + 2901z24 + 377z26 + 29z28 + z30

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

Determinant and Signature: {3, 22}

Jones Polynomial: q15 + q17 - q32

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: {85, 680}

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=22 is the signature of T(16,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
0123456789101112131415161718192021χ
65                     1-1
63                     1-1
61                   11 0
59                 1  1 0
57                 11   0
55               11     0
53             1  1     0
51             11       0
49           11         0
47         1  1         0
45         11           0
43       11             0
41     1  1             0
39     11               0
37   11                 0
35    1                 1
33  1                   1
311                     1
291                     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[16, 3]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[16, 3]]
Out[3]=   
32
In[4]:=
PD[TorusKnot[16, 3]]
Out[4]=   
PD[X[41, 63, 42, 62], X[20, 64, 21, 63], X[21, 43, 22, 42], X[64, 44, 1, 43], 
 
>   X[1, 23, 2, 22], X[44, 24, 45, 23], X[45, 3, 46, 2], X[24, 4, 25, 3], 
 
>   X[25, 47, 26, 46], X[4, 48, 5, 47], X[5, 27, 6, 26], X[48, 28, 49, 27], 
 
>   X[49, 7, 50, 6], X[28, 8, 29, 7], X[29, 51, 30, 50], X[8, 52, 9, 51], 
 
>   X[9, 31, 10, 30], X[52, 32, 53, 31], X[53, 11, 54, 10], X[32, 12, 33, 11], 
 
>   X[33, 55, 34, 54], X[12, 56, 13, 55], X[13, 35, 14, 34], X[56, 36, 57, 35], 
 
>   X[57, 15, 58, 14], X[36, 16, 37, 15], X[37, 59, 38, 58], X[16, 60, 17, 59], 
 
>   X[17, 39, 18, 38], X[60, 40, 61, 39], X[61, 19, 62, 18], X[40, 20, 41, 19]]
In[5]:=
GaussCode[TorusKnot[16, 3]]
Out[5]=   
GaussCode[-5, 7, 8, -10, -11, 13, 14, -16, -17, 19, 20, -22, -23, 25, 26, -28, 
 
>   -29, 31, 32, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 17, 18, -20, -21, 23, 
 
>   24, -26, -27, 29, 30, -32, -1, 3, 4, -6, -7, 9, 10, -12, -13, 15, 16, -18, 
 
>   -19, 21, 22, -24, -25, 27, 28, -30, -31, 1, 2, -4]
In[6]:=
BR[TorusKnot[16, 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}]
In[7]:=
alex = Alexander[TorusKnot[16, 3]][t]
Out[7]=   
     -15    -14    -12    -11    -9    -8    -6    -5    -3    -2    2    3
1 + t    - t    + t    - t    + t   - t   + t   - t   + t   - t   - t  + t  - 
 
     5    6    8    9    11    12    14    15
>   t  + t  - t  + t  - t   + t   - t   + t
In[8]:=
Conway[TorusKnot[16, 3]][z]
Out[8]=   
        2         4          6          8           10           12
1 + 85 z  + 1785 z  + 14637 z  + 62305 z  + 158389 z   + 260729 z   + 
 
            14           16           18          20          22         24
>   292077 z   + 229517 z   + 128593 z   + 51589 z   + 14697 z   + 2901 z   + 
 
         26       28    30
>   377 z   + 29 z   + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{}
In[10]:=
{KnotDet[TorusKnot[16, 3]], KnotSignature[TorusKnot[16, 3]]}
Out[10]=   
{3, 22}
In[11]:=
J=Jones[TorusKnot[16, 3]][q]
Out[11]=   
 15    17    32
q   + q   - q
In[12]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[12]=   
{}
In[13]:=
A2Invariant[TorusKnot[16, 3]][q]
Out[13]=   
NotAvailable
In[14]:=
Kauffman[TorusKnot[16, 3]][a, z]
Out[14]=   
NotAvailable
In[15]:=
{Vassiliev[2][TorusKnot[16, 3]], Vassiliev[3][TorusKnot[16, 3]]}
Out[15]=   
{85, 680}
In[16]:=
Kh[TorusKnot[16, 3]][q, t]
Out[16]=   
 29    31    33  2    37  3    35  4    37  4    39  5    41  5    39  6
q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 
 
     43  7    41  8    43  8    45  9    47  9    45  10    49  11    47  12
>   q   t  + q   t  + q   t  + q   t  + q   t  + q   t   + q   t   + q   t   + 
 
     49  12    51  13    53  13    51  14    55  15    53  16    55  16
>   q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     57  17    59  17    57  18    61  19    59  20    61  20    63  21
>   q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     65  21
>   q   t


Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(16,3)
T(8,5)
T(8,5)
T(11,4)
T(11,4)