© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(16,3)
T(16,3)
T(33,2)
T(33,2)
T(11,4)
TubePlot
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   The 33-Crossing Torus Knot T(11,4)

Visit T(11,4)'s page at Knotilus!

Acknowledgement

PD Presentation: X3,53,4,52 X20,54,21,53 X37,55,38,54 X21,5,22,4 X38,6,39,5 X55,7,56,6 X39,23,40,22 X56,24,57,23 X7,25,8,24 X57,41,58,40 X8,42,9,41 X25,43,26,42 X9,59,10,58 X26,60,27,59 X43,61,44,60 X27,11,28,10 X44,12,45,11 X61,13,62,12 X45,29,46,28 X62,30,63,29 X13,31,14,30 X63,47,64,46 X14,48,15,47 X31,49,32,48 X15,65,16,64 X32,66,33,65 X49,1,50,66 X33,17,34,16 X50,18,51,17 X1,19,2,18 X51,35,52,34 X2,36,3,35 X19,37,20,36

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

Braid Representative:    

Alexander Polynomial: t-15 - t-14 + t-11 - t-10 + t-7 - t-6 + t-4 - t-2 + 1 - t2 + t4 - t6 + t7 - t10 + t11 - t14 + t15

Conway Polynomial: 1 + 75z2 + 1510z4 + 12825z6 + 56577z8 + 148070z10 + 249288z12 + 283951z14 + 225760z16 + 127470z18 + 51380z20 + 14675z22 + 2900z24 + 377z26 + 29z28 + z30

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

Determinant and Signature: {11, 22}

Jones Polynomial: q15 + q17 + q19 - q20 + q21 - q22 + q23 - q24 + q25 - 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: {75, 550}

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(11,4). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
0123456789101112131415161718192021χ
63                    110
61                      0
59                  121 0
57                12    -1
55                 21   -1
53               32     -1
51            12  1     0
49           1 12       0
47           22         0
45         21 1         0
43       1  1           0
41     1 12             0
39     11               0
37   11 1               1
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[11, 4]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[11, 4]]
Out[3]=   
33
In[4]:=
PD[TorusKnot[11, 4]]
Out[4]=   
PD[X[3, 53, 4, 52], X[20, 54, 21, 53], X[37, 55, 38, 54], X[21, 5, 22, 4], 
 
>   X[38, 6, 39, 5], X[55, 7, 56, 6], X[39, 23, 40, 22], X[56, 24, 57, 23], 
 
>   X[7, 25, 8, 24], X[57, 41, 58, 40], X[8, 42, 9, 41], X[25, 43, 26, 42], 
 
>   X[9, 59, 10, 58], X[26, 60, 27, 59], X[43, 61, 44, 60], X[27, 11, 28, 10], 
 
>   X[44, 12, 45, 11], X[61, 13, 62, 12], X[45, 29, 46, 28], X[62, 30, 63, 29], 
 
>   X[13, 31, 14, 30], X[63, 47, 64, 46], X[14, 48, 15, 47], X[31, 49, 32, 48], 
 
>   X[15, 65, 16, 64], X[32, 66, 33, 65], X[49, 1, 50, 66], X[33, 17, 34, 16], 
 
>   X[50, 18, 51, 17], X[1, 19, 2, 18], X[51, 35, 52, 34], X[2, 36, 3, 35], 
 
>   X[19, 37, 20, 36]]
In[5]:=
GaussCode[TorusKnot[11, 4]]
Out[5]=   
GaussCode[-30, -32, -1, 4, 5, 6, -9, -11, -13, 16, 17, 18, -21, -23, -25, 28, 
 
>   29, 30, -33, -2, -4, 7, 8, 9, -12, -14, -16, 19, 20, 21, -24, -26, -28, 31, 
 
>   32, 33, -3, -5, -7, 10, 11, 12, -15, -17, -19, 22, 23, 24, -27, -29, -31, 
 
>   1, 2, 3, -6, -8, -10, 13, 14, 15, -18, -20, -22, 25, 26, 27]
In[6]:=
BR[TorusKnot[11, 4]]
Out[6]=   
BR[4, {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 
 
>    1, 2, 3, 1, 2, 3, 1, 2, 3}]
In[7]:=
alex = Alexander[TorusKnot[11, 4]][t]
Out[7]=   
     -15    -14    -11    -10    -7    -6    -4    -2    2    4    6    7
1 + t    - t    + t    - t    + t   - t   + t   - t   - t  + t  - t  + t  - 
 
     10    11    14    15
>   t   + t   - t   + t
In[8]:=
Conway[TorusKnot[11, 4]][z]
Out[8]=   
        2         4          6          8           10           12
1 + 75 z  + 1510 z  + 12825 z  + 56577 z  + 148070 z   + 249288 z   + 
 
            14           16           18          20          22         24
>   283951 z   + 225760 z   + 127470 z   + 51380 z   + 14675 z   + 2900 z   + 
 
         26       28    30
>   377 z   + 29 z   + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{}
In[10]:=
{KnotDet[TorusKnot[11, 4]], KnotSignature[TorusKnot[11, 4]]}
Out[10]=   
{11, 22}
In[11]:=
J=Jones[TorusKnot[11, 4]][q]
Out[11]=   
 15    17    19    20    21    22    23    24    25    26    28
q   + q   + q   - q   + q   - q   + q   - q   + q   - q   - q
In[12]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[12]=   
{}
In[13]:=
A2Invariant[TorusKnot[11, 4]][q]
Out[13]=   
NotAvailable
In[14]:=
Kauffman[TorusKnot[11, 4]][a, z]
Out[14]=   
NotAvailable
In[15]:=
{Vassiliev[2][TorusKnot[11, 4]], Vassiliev[3][TorusKnot[11, 4]]}
Out[15]=   
{75, 550}
In[16]:=
Kh[TorusKnot[11, 4]][q, t]
Out[16]=   
 29    31    33  2    37  3    35  4    37  4    39  5    41  5    37  6
q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 
 
     39  6    41  7    43  7      41  8      45  9    43  10    45  10
>   q   t  + q   t  + q   t  + 2 q   t  + 2 q   t  + q   t   + q   t   + 
 
       47  11    49  11    45  12      47  12    51  12    49  13      51  13
>   2 q   t   + q   t   + q   t   + 2 q   t   + q   t   + q   t   + 2 q   t   + 
 
       49  14      53  15    51  16      53  16    57  16      55  17
>   2 q   t   + 3 q   t   + q   t   + 2 q   t   + q   t   + 2 q   t   + 
 
       57  17    55  18    59  18      59  19    59  20    63  20    63  21
>   2 q   t   + q   t   + q   t   + 2 q   t   + q   t   + q   t   + q   t


Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(11,4)
T(16,3)
T(16,3)
T(33,2)
T(33,2)