© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(27,2)
T(27,2)
T(14,3)
T(14,3)
T(7,5)
TubePlot
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   The 28-Crossing Torus Knot T(7,5)

Visit T(7,5)'s page at Knotilus!

Acknowledgement

PD Presentation: X25,3,26,2 X48,4,49,3 X15,5,16,4 X38,6,39,5 X49,27,50,26 X16,28,17,27 X39,29,40,28 X6,30,7,29 X17,51,18,50 X40,52,41,51 X7,53,8,52 X30,54,31,53 X41,19,42,18 X8,20,9,19 X31,21,32,20 X54,22,55,21 X9,43,10,42 X32,44,33,43 X55,45,56,44 X22,46,23,45 X33,11,34,10 X56,12,1,11 X23,13,24,12 X46,14,47,13 X1,35,2,34 X24,36,25,35 X47,37,48,36 X14,38,15,37

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

Braid Representative:    

Alexander Polynomial: t-12 - t-11 + t-7 - t-6 + t-5 - t-4 + t-2 - t-1 + 1 - t + t2 - t4 + t5 - t6 + t7 - t11 + t12

Conway Polynomial: 1 + 48z2 + 628z4 + 3498z6 + 10032z8 + 16511z10 + 16757z12 + 10949z14 + 4692z16 + 1311z18 + 230z20 + 23z22 + z24

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

Determinant and Signature: {1, 16}

Jones Polynomial: q12 + q14 + q16 - q20 - q22

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: {48, 280}

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(7,5). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
01234567891011121314151617χ
51                110
49                  0
47              121 0
45            12    -1
43             21   -1
41           32     -1
39         2  1     -1
37       1 12       0
35     1 12         0
33     11 1         1
31   11 1           1
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[7, 5]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[7, 5]]
Out[3]=   
28
In[4]:=
PD[TorusKnot[7, 5]]
Out[4]=   
PD[X[25, 3, 26, 2], X[48, 4, 49, 3], X[15, 5, 16, 4], X[38, 6, 39, 5], 
 
>   X[49, 27, 50, 26], X[16, 28, 17, 27], X[39, 29, 40, 28], X[6, 30, 7, 29], 
 
>   X[17, 51, 18, 50], X[40, 52, 41, 51], X[7, 53, 8, 52], X[30, 54, 31, 53], 
 
>   X[41, 19, 42, 18], X[8, 20, 9, 19], X[31, 21, 32, 20], X[54, 22, 55, 21], 
 
>   X[9, 43, 10, 42], X[32, 44, 33, 43], X[55, 45, 56, 44], X[22, 46, 23, 45], 
 
>   X[33, 11, 34, 10], X[56, 12, 1, 11], X[23, 13, 24, 12], X[46, 14, 47, 13], 
 
>   X[1, 35, 2, 34], X[24, 36, 25, 35], X[47, 37, 48, 36], X[14, 38, 15, 37]]
In[5]:=
GaussCode[TorusKnot[7, 5]]
Out[5]=   
GaussCode[-25, 1, 2, 3, 4, -8, -11, -14, -17, 21, 22, 23, 24, -28, -3, -6, -9, 
 
>   13, 14, 15, 16, -20, -23, -26, -1, 5, 6, 7, 8, -12, -15, -18, -21, 25, 26, 
 
>   27, 28, -4, -7, -10, -13, 17, 18, 19, 20, -24, -27, -2, -5, 9, 10, 11, 12, 
 
>   -16, -19, -22]
In[6]:=
BR[TorusKnot[7, 5]]
Out[6]=   
BR[5, {1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 
 
>    1, 2, 3, 4}]
In[7]:=
alex = Alexander[TorusKnot[7, 5]][t]
Out[7]=   
     -12    -11    -7    -6    -5    -4    -2   1        2    4    5    6
1 + t    - t    + t   - t   + t   - t   + t   - - - t + t  - t  + t  - t  + 
                                                t
 
     7    11    12
>   t  - t   + t
In[8]:=
Conway[TorusKnot[7, 5]][z]
Out[8]=   
        2        4         6          8          10          12          14
1 + 48 z  + 628 z  + 3498 z  + 10032 z  + 16511 z   + 16757 z   + 10949 z   + 
 
          16         18        20       22    24
>   4692 z   + 1311 z   + 230 z   + 23 z   + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{}
In[10]:=
{KnotDet[TorusKnot[7, 5]], KnotSignature[TorusKnot[7, 5]]}
Out[10]=   
{1, 16}
In[11]:=
J=Jones[TorusKnot[7, 5]][q]
Out[11]=   
 12    14    16    20    22
q   + q   + q   - q   - q
In[12]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[12]=   
{}
In[13]:=
A2Invariant[TorusKnot[7, 5]][q]
Out[13]=   
NotAvailable
In[14]:=
Kauffman[TorusKnot[7, 5]][a, z]
Out[14]=   
NotAvailable
In[15]:=
{Vassiliev[2][TorusKnot[7, 5]], Vassiliev[3][TorusKnot[7, 5]]}
Out[15]=   
{48, 280}
In[16]:=
Kh[TorusKnot[7, 5]][q, t]
Out[16]=   
 23    25    27  2    31  3    29  4    31  4    33  5    35  5    31  6
q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 
 
     33  6    35  7    37  7    33  8      35  8    37  9      39  9
>   q   t  + q   t  + q   t  + q   t  + 2 q   t  + q   t  + 2 q   t  + 
 
       37  10      41  11    39  12      41  12    45  12      43  13
>   2 q   t   + 3 q   t   + q   t   + 2 q   t   + q   t   + 2 q   t   + 
 
       45  13    43  14    47  14      47  15    47  16    51  16    51  17
>   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(7,5)
T(27,2)
T(27,2)
T(14,3)
T(14,3)