© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(17,3)
T(17,3)
T(35,2)
T(35,2)
T(7,6)
TubePlot
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   The 35-Crossing Torus Knot T(7,6)

Visit T(7,6)'s page at Knotilus!

Acknowledgement

PD Presentation: X53,65,54,64 X42,66,43,65 X31,67,32,66 X20,68,21,67 X9,69,10,68 X43,55,44,54 X32,56,33,55 X21,57,22,56 X10,58,11,57 X69,59,70,58 X33,45,34,44 X22,46,23,45 X11,47,12,46 X70,48,1,47 X59,49,60,48 X23,35,24,34 X12,36,13,35 X1,37,2,36 X60,38,61,37 X49,39,50,38 X13,25,14,24 X2,26,3,25 X61,27,62,26 X50,28,51,27 X39,29,40,28 X3,15,4,14 X62,16,63,15 X51,17,52,16 X40,18,41,17 X29,19,30,18 X63,5,64,4 X52,6,53,5 X41,7,42,6 X30,8,31,7 X19,9,20,8

Gauss Code: {-18, -22, -26, 31, 32, 33, 34, 35, -5, -9, -13, -17, -21, 26, 27, 28, 29, 30, -35, -4, -8, -12, -16, 21, 22, 23, 24, 25, -30, -34, -3, -7, -11, 16, 17, 18, 19, 20, -25, -29, -33, -2, -6, 11, 12, 13, 14, 15, -20, -24, -28, -32, -1, 6, 7, 8, 9, 10, -15, -19, -23, -27, -31, 1, 2, 3, 4, 5, -10, -14}

Braid Representative:    

Alexander Polynomial: t-15 - t-14 + t-9 - t-7 + t-3 - 1 + t3 - t7 + t9 - t14 + t15

Conway Polynomial: 1 + 70z2 + 1365z4 + 11649z6 + 52844z8 + 142208z10 + 244074z12 + 281144z14 + 224826z16 + 127282z18 + 51359z20 + 14674z22 + 2900z24 + 377z26 + 29z28 + z30

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

Determinant and Signature: {7, 18}

Jones Polynomial: q15 + q17 + q19 + q21 - q22 - q24 - 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: {70, 490}

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=18 is the signature of T(7,6). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
012345678910111213141516171819χ
57                1  10
55                11  0
53              12 11 -1
51            11 21   -1
49             31 1   -1
47           31 1     -1
45         2 12       -1
43       1 12         0
41     1 12 1         1
39     11 1           1
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[7, 6]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[7, 6]]
Out[3]=   
35
In[4]:=
PD[TorusKnot[7, 6]]
Out[4]=   
PD[X[53, 65, 54, 64], X[42, 66, 43, 65], X[31, 67, 32, 66], X[20, 68, 21, 67], 
 
>   X[9, 69, 10, 68], X[43, 55, 44, 54], X[32, 56, 33, 55], X[21, 57, 22, 56], 
 
>   X[10, 58, 11, 57], X[69, 59, 70, 58], X[33, 45, 34, 44], X[22, 46, 23, 45], 
 
>   X[11, 47, 12, 46], X[70, 48, 1, 47], X[59, 49, 60, 48], X[23, 35, 24, 34], 
 
>   X[12, 36, 13, 35], X[1, 37, 2, 36], X[60, 38, 61, 37], X[49, 39, 50, 38], 
 
>   X[13, 25, 14, 24], X[2, 26, 3, 25], X[61, 27, 62, 26], X[50, 28, 51, 27], 
 
>   X[39, 29, 40, 28], X[3, 15, 4, 14], X[62, 16, 63, 15], X[51, 17, 52, 16], 
 
>   X[40, 18, 41, 17], X[29, 19, 30, 18], X[63, 5, 64, 4], X[52, 6, 53, 5], 
 
>   X[41, 7, 42, 6], X[30, 8, 31, 7], X[19, 9, 20, 8]]
In[5]:=
GaussCode[TorusKnot[7, 6]]
Out[5]=   
GaussCode[-18, -22, -26, 31, 32, 33, 34, 35, -5, -9, -13, -17, -21, 26, 27, 28, 
 
>   29, 30, -35, -4, -8, -12, -16, 21, 22, 23, 24, 25, -30, -34, -3, -7, -11, 
 
>   16, 17, 18, 19, 20, -25, -29, -33, -2, -6, 11, 12, 13, 14, 15, -20, -24, 
 
>   -28, -32, -1, 6, 7, 8, 9, 10, -15, -19, -23, -27, -31, 1, 2, 3, 4, 5, -10, 
 
>   -14]
In[6]:=
BR[TorusKnot[7, 6]]
Out[6]=   
BR[6, {1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 
 
>    5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5}]
In[7]:=
alex = Alexander[TorusKnot[7, 6]][t]
Out[7]=   
      -15    -14    -9    -7    -3    3    7    9    14    15
-1 + t    - t    + t   - t   + t   + t  - t  + t  - t   + t
In[8]:=
Conway[TorusKnot[7, 6]][z]
Out[8]=   
        2         4          6          8           10           12
1 + 70 z  + 1365 z  + 11649 z  + 52844 z  + 142208 z   + 244074 z   + 
 
            14           16           18          20          22         24
>   281144 z   + 224826 z   + 127282 z   + 51359 z   + 14674 z   + 2900 z   + 
 
         26       28    30
>   377 z   + 29 z   + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{}
In[10]:=
{KnotDet[TorusKnot[7, 6]], KnotSignature[TorusKnot[7, 6]]}
Out[10]=   
{7, 18}
In[11]:=
J=Jones[TorusKnot[7, 6]][q]
Out[11]=   
 15    17    19    21    22    24    26
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[7, 6]][q]
Out[13]=   
NotAvailable
In[14]:=
Kauffman[TorusKnot[7, 6]][a, z]
Out[14]=   
NotAvailable
In[15]:=
{Vassiliev[2][TorusKnot[7, 6]], Vassiliev[3][TorusKnot[7, 6]]}
Out[15]=   
{70, 490}
In[16]:=
Kh[TorusKnot[7, 6]][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    39  8      41  8    43  9      45  9
>   q   t  + q   t  + q   t  + q   t  + 2 q   t  + q   t  + 2 q   t  + 
 
     41  10      43  10    45  11      47  11      45  12    47  12    51  12
>   q   t   + 2 q   t   + q   t   + 3 q   t   + 2 q   t   + q   t   + q   t   + 
 
       49  13    51  13    47  14    49  14    53  14      51  15      53  15
>   3 q   t   + q   t   + q   t   + q   t   + q   t   + 2 q   t   + 2 q   t   + 
 
     49  16    51  16    55  16    57  16    53  17    55  17    53  18
>   q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     57  19
>   q   t


Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(7,6)
T(17,3)
T(17,3)
T(35,2)
T(35,2)