© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(7,4)
T(7,4) 		T(11,3)
T(11,3)
T(21,2)
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   The 21-Crossing Torus Knot T(21,2)

Visit T(21,2)'s page at Knotilus!

Acknowledgement

PD Presentation: X11,33,12,32 X33,13,34,12 X13,35,14,34 X35,15,36,14 X15,37,16,36 X37,17,38,16 X17,39,18,38 X39,19,40,18 X19,41,20,40 X41,21,42,20 X21,1,22,42 X1,23,2,22 X23,3,24,2 X3,25,4,24 X25,5,26,4 X5,27,6,26 X27,7,28,6 X7,29,8,28 X29,9,30,8 X9,31,10,30 X31,11,32,10

Gauss Code: {-12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11}

Braid Representative:    

Alexander Polynomial: t-10 - t-9 + 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 - t9 + t10

Conway Polynomial: 1 + 55z2 + 495z4 + 1716z6 + 3003z8 + 3003z10 + 1820z12 + 680z14 + 153z16 + 19z18 + z20

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

Determinant and Signature: {21, 20}

Jones Polynomial: q10 + q12 - q13 + q14 - q15 + q16 - q17 + q18 - q19 + q20 - q21 + q22 - q23 + q24 - q25 + q26 - q27 + q28 - q29 + q30 - q31

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: {55, 385}

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(21,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 0123456789101112131415161718192021χ
63                     1-1
61                      0
59                   11 0
57                      0
55                 11   0
53                      0
51               11     0
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  1                   1
211                     1
191                     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[21, 2]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[21, 2]]
Out[3]=   
21
In[4]:=
PD[TorusKnot[21, 2]]
Out[4]=   
PD[X[11, 33, 12, 32], X[33, 13, 34, 12], X[13, 35, 14, 34], X[35, 15, 36, 14], 
 
>   X[15, 37, 16, 36], X[37, 17, 38, 16], X[17, 39, 18, 38], X[39, 19, 40, 18], 
 
>   X[19, 41, 20, 40], X[41, 21, 42, 20], X[21, 1, 22, 42], X[1, 23, 2, 22], 
 
>   X[23, 3, 24, 2], X[3, 25, 4, 24], X[25, 5, 26, 4], X[5, 27, 6, 26], 
 
>   X[27, 7, 28, 6], X[7, 29, 8, 28], X[29, 9, 30, 8], X[9, 31, 10, 30], 
 
>   X[31, 11, 32, 10]]
In[5]:=
GaussCode[TorusKnot[21, 2]]
Out[5]=   
GaussCode[-12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -1, 2, -3, 4, -5, 6, -7, 
 
>   8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 1, -2, 3, -4, 
 
>   5, -6, 7, -8, 9, -10, 11]
In[6]:=
BR[TorusKnot[21, 2]]
Out[6]=   
BR[2, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}]
In[7]:=
alex = Alexander[TorusKnot[21, 2]][t]
Out[7]=   
     -10    -9    -8    -7    -6    -5    -4    -3    -2   1        2    3
1 + t    - t   + t   - t   + t   - t   + t   - t   + t   - - - t + t  - t  + 
                                                           t
 
     4    5    6    7    8    9    10
>   t  - t  + t  - t  + t  - t  + t
In[8]:=
Conway[TorusKnot[21, 2]][z]
Out[8]=   
        2        4         6         8         10         12        14
1 + 55 z  + 495 z  + 1716 z  + 3003 z  + 3003 z   + 1820 z   + 680 z   + 
 
         16       18    20
>   153 z   + 19 z   + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{}
In[10]:=
{KnotDet[TorusKnot[21, 2]], KnotSignature[TorusKnot[21, 2]]}
Out[10]=   
{21, 20}
In[11]:=
J=Jones[TorusKnot[21, 2]][q]
Out[11]=   
 10    12    13    14    15    16    17    18    19    20    21    22    23
q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + 
 
     24    25    26    27    28    29    30    31
>   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[21, 2]][q]
Out[13]=   
NotAvailable
In[14]:=
Kauffman[TorusKnot[21, 2]][a, z]
Out[14]=   
NotAvailable
In[15]:=
{Vassiliev[2][TorusKnot[21, 2]], Vassiliev[3][TorusKnot[21, 2]]}
Out[15]=   
{55, 385}
In[16]:=
Kh[TorusKnot[21, 2]][q, t]
Out[16]=   
 19    21    23  2    27  3    27  4    31  5    31  6    35  7    35  8
q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 
 
     39  9    39  10    43  11    43  12    47  13    47  14    51  15
>   q   t  + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     51  16    55  17    55  18    59  19    59  20    63  21
>   q   t   + q   t   + q   t   + q   t   + q   t   + q   t

Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(21,2)
T(7,4)
T(7,4) 		T(11,3)
T(11,3)