© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(11,3)
T(11,3)
T(6,5)
T(6,5)
T(23,2)
TubePlot
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   The 23-Crossing Torus Knot T(23,2)

Visit T(23,2)'s page at Knotilus!

Acknowledgement

PD Presentation: X9,33,10,32 X33,11,34,10 X11,35,12,34 X35,13,36,12 X13,37,14,36 X37,15,38,14 X15,39,16,38 X39,17,40,16 X17,41,18,40 X41,19,42,18 X19,43,20,42 X43,21,44,20 X21,45,22,44 X45,23,46,22 X23,1,24,46 X1,25,2,24 X25,3,26,2 X3,27,4,26 X27,5,28,4 X5,29,6,28 X29,7,30,6 X7,31,8,30 X31,9,32,8

Gauss Code: {-16, 17, -18, 19, -20, 21, -22, 23, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15}

Braid Representative:    

Alexander Polynomial: t-11 - 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 + t11

Conway Polynomial: 1 + 66z2 + 715z4 + 3003z6 + 6435z8 + 8008z10 + 6188z12 + 3060z14 + 969z16 + 190z18 + 21z20 + z22

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

Determinant and Signature: {23, 22}

Jones Polynomial: q11 + q13 - q14 + q15 - q16 + q17 - q18 + q19 - q20 + q21 - q22 + q23 - q24 + q25 - q26 + q27 - q28 + q29 - q30 + q31 - q32 + q33 - 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: {66, 506}

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(23,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
01234567891011121314151617181920212223χ
69                       1-1
67                        0
65                     11 0
63                        0
61                   11   0
59                        0
57                 11     0
55                        0
53               11       0
51                        0
49             11         0
47                        0
45           11           0
43                        0
41         11             0
39                        0
37       11               0
35                        0
33     11                 0
31                        0
29   11                   0
27                        0
25  1                     1
231                       1
211                       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[23, 2]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[23, 2]]
Out[3]=   
23
In[4]:=
PD[TorusKnot[23, 2]]
Out[4]=   
PD[X[9, 33, 10, 32], X[33, 11, 34, 10], X[11, 35, 12, 34], X[35, 13, 36, 12], 
 
>   X[13, 37, 14, 36], X[37, 15, 38, 14], X[15, 39, 16, 38], X[39, 17, 40, 16], 
 
>   X[17, 41, 18, 40], X[41, 19, 42, 18], X[19, 43, 20, 42], X[43, 21, 44, 20], 
 
>   X[21, 45, 22, 44], X[45, 23, 46, 22], X[23, 1, 24, 46], X[1, 25, 2, 24], 
 
>   X[25, 3, 26, 2], X[3, 27, 4, 26], X[27, 5, 28, 4], X[5, 29, 6, 28], 
 
>   X[29, 7, 30, 6], X[7, 31, 8, 30], X[31, 9, 32, 8]]
In[5]:=
GaussCode[TorusKnot[23, 2]]
Out[5]=   
GaussCode[-16, 17, -18, 19, -20, 21, -22, 23, -1, 2, -3, 4, -5, 6, -7, 8, -9, 
 
>   10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 1, -2, 3, 
 
>   -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15]
In[6]:=
BR[TorusKnot[23, 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, 1, 1}]
In[7]:=
alex = Alexander[TorusKnot[23, 2]][t]
Out[7]=   
      -11    -10    -9    -8    -7    -6    -5    -4    -3    -2   1        2
-1 + t    - t    + t   - t   + t   - t   + t   - t   + t   - t   + - + t - t  + 
                                                                   t
 
     3    4    5    6    7    8    9    10    11
>   t  - t  + t  - t  + t  - t  + t  - t   + t
In[8]:=
Conway[TorusKnot[23, 2]][z]
Out[8]=   
        2        4         6         8         10         12         14
1 + 66 z  + 715 z  + 3003 z  + 6435 z  + 8008 z   + 6188 z   + 3060 z   + 
 
         16        18       20    22
>   969 z   + 190 z   + 21 z   + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{}
In[10]:=
{KnotDet[TorusKnot[23, 2]], KnotSignature[TorusKnot[23, 2]]}
Out[10]=   
{23, 22}
In[11]:=
J=Jones[TorusKnot[23, 2]][q]
Out[11]=   
 11    13    14    15    16    17    18    19    20    21    22    23    24
q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + 
 
     25    26    27    28    29    30    31    32    33    34
>   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[23, 2]][q]
Out[13]=   
NotAvailable
In[14]:=
Kauffman[TorusKnot[23, 2]][a, z]
Out[14]=   
NotAvailable
In[15]:=
{Vassiliev[2][TorusKnot[23, 2]], Vassiliev[3][TorusKnot[23, 2]]}
Out[15]=   
{66, 506}
In[16]:=
Kh[TorusKnot[23, 2]][q, t]
Out[16]=   
 21    23    25  2    29  3    29  4    33  5    33  6    37  7    37  8
q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 
 
     41  9    41  10    45  11    45  12    49  13    49  14    53  15
>   q   t  + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     53  16    57  17    57  18    61  19    61  20    65  21    65  22
>   q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     69  23
>   q   t


Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(23,2)
T(11,3)
T(11,3)
T(6,5)
T(6,5)