© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(25,2)
T(25,2) 		T(9,4)
T(9,4)
T(13,3)
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   The 26-Crossing Torus Knot T(13,3)

Visit T(13,3)'s page at Knotilus!

Acknowledgement

PD Presentation: X38,4,39,3 X21,5,22,4 X22,40,23,39 X5,41,6,40 X6,24,7,23 X41,25,42,24 X42,8,43,7 X25,9,26,8 X26,44,27,43 X9,45,10,44 X10,28,11,27 X45,29,46,28 X46,12,47,11 X29,13,30,12 X30,48,31,47 X13,49,14,48 X14,32,15,31 X49,33,50,32 X50,16,51,15 X33,17,34,16 X34,52,35,51 X17,1,18,52 X18,36,19,35 X1,37,2,36 X2,20,3,19 X37,21,38,20

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

Braid Representative:    

Alexander Polynomial: 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

Conway Polynomial: 1 + 56z2 + 770z4 + 4081z6 + 11033z8 + 17391z10 + 17187z12 + 11067z14 + 4709z16 + 1312z18 + 230z20 + 23z22 + z24

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

Determinant and Signature: {1, 16}

Jones Polynomial: q12 + q14 - q26

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: {56, 364}

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=16 is the signature of T(13,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 01234567891011121314151617χ
53                 1-1
51                 1-1
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         0
33     11           0
31   11             0
29    1             1
27  1               1
251                 1
231                 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[13, 3]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[13, 3]]
Out[3]=   
26
In[4]:=
PD[TorusKnot[13, 3]]
Out[4]=   
PD[X[38, 4, 39, 3], X[21, 5, 22, 4], X[22, 40, 23, 39], X[5, 41, 6, 40], 
 
>   X[6, 24, 7, 23], X[41, 25, 42, 24], X[42, 8, 43, 7], X[25, 9, 26, 8], 
 
>   X[26, 44, 27, 43], X[9, 45, 10, 44], X[10, 28, 11, 27], X[45, 29, 46, 28], 
 
>   X[46, 12, 47, 11], X[29, 13, 30, 12], X[30, 48, 31, 47], X[13, 49, 14, 48], 
 
>   X[14, 32, 15, 31], X[49, 33, 50, 32], X[50, 16, 51, 15], X[33, 17, 34, 16], 
 
>   X[34, 52, 35, 51], X[17, 1, 18, 52], X[18, 36, 19, 35], X[1, 37, 2, 36], 
 
>   X[2, 20, 3, 19], X[37, 21, 38, 20]]
In[5]:=
GaussCode[TorusKnot[13, 3]]
Out[5]=   
GaussCode[-24, -25, 1, 2, -4, -5, 7, 8, -10, -11, 13, 14, -16, -17, 19, 20, 
 
>   -22, -23, 25, 26, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 17, 18, -20, -21, 
 
>   23, 24, -26, -1, 3, 4, -6, -7, 9, 10, -12, -13, 15, 16, -18, -19, 21, 22]
In[6]:=
BR[TorusKnot[13, 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}]
In[7]:=
alex = Alexander[TorusKnot[13, 3]][t]
Out[7]=   
     -12    -11    -9    -8    -6    -5    -3    -2    2    3    5    6    8
1 + t    - t    + t   - t   + t   - t   + t   - t   - t  + t  - t  + t  - t  + 
 
     9    11    12
>   t  - t   + t
In[8]:=
Conway[TorusKnot[13, 3]][z]
Out[8]=   
        2        4         6          8          10          12          14
1 + 56 z  + 770 z  + 4081 z  + 11033 z  + 17391 z   + 17187 z   + 11067 z   + 
 
          16         18        20       22    24
>   4709 z   + 1312 z   + 230 z   + 23 z   + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{}
In[10]:=
{KnotDet[TorusKnot[13, 3]], KnotSignature[TorusKnot[13, 3]]}
Out[10]=   
{1, 16}
In[11]:=
J=Jones[TorusKnot[13, 3]][q]
Out[11]=   
 12    14    26
q   + q   - q
In[12]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[12]=   
{}
In[13]:=
A2Invariant[TorusKnot[13, 3]][q]
Out[13]=   
NotAvailable
In[14]:=
Kauffman[TorusKnot[13, 3]][a, z]
Out[14]=   
NotAvailable
In[15]:=
{Vassiliev[2][TorusKnot[13, 3]], Vassiliev[3][TorusKnot[13, 3]]}
Out[15]=   
{56, 364}
In[16]:=
Kh[TorusKnot[13, 3]][q, t]
Out[16]=   
 23    25    27  2    31  3    29  4    31  4    33  5    35  5    33  6
q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 
 
     37  7    35  8    37  8    39  9    41  9    39  10    43  11    41  12
>   q   t  + q   t  + q   t  + q   t  + q   t  + q   t   + q   t   + q   t   + 
 
     43  12    45  13    47  13    45  14    49  15    47  16    49  16
>   q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     51  17    53  17
>   q   t   + q   t

Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(13,3)
T(25,2)
T(25,2) 		T(9,4)
T(9,4)