PD Presentation: |
X54,16,55,15 X29,17,30,16 X4,18,5,17 X43,19,44,18 X30,56,31,55 X5,57,6,56 X44,58,45,57 X19,59,20,58 X6,32,7,31 X45,33,46,32 X20,34,21,33 X59,35,60,34 X46,8,47,7 X21,9,22,8 X60,10,61,9 X35,11,36,10 X22,48,23,47 X61,49,62,48 X36,50,37,49 X11,51,12,50 X62,24,63,23 X37,25,38,24 X12,26,13,25 X51,27,52,26 X38,64,39,63 X13,1,14,64 X52,2,53,1 X27,3,28,2 X14,40,15,39 X53,41,54,40 X28,42,29,41 X3,43,4,42 |
Gauss Code: |
{27, 28, -32, -3, -6, -9, 13, 14, 15, 16, -20, -23, -26, -29, 1, 2, 3, 4, -8, -11, -14, -17, 21, 22, 23, 24, -28, -31, -2, -5, 9, 10, 11, 12, -16, -19, -22, -25, 29, 30, 31, 32, -4, -7, -10, -13, 17, 18, 19, 20, -24, -27, -30, -1, 5, 6, 7, 8, -12, -15, -18, -21, 25, 26} |
In[1]:= |
<< KnotTheory` |
Loading KnotTheory` (version of August 30, 2005, 10:15:35)... |
In[2]:= | TubePlot[TorusKnot[8, 5]] |
|  |
Out[2]= | -Graphics- |
In[3]:= | Crossings[TorusKnot[8, 5]] |
Out[3]= | 32 |
In[4]:= | PD[TorusKnot[8, 5]] |
Out[4]= | PD[X[54, 16, 55, 15], X[29, 17, 30, 16], X[4, 18, 5, 17], X[43, 19, 44, 18],
> X[30, 56, 31, 55], X[5, 57, 6, 56], X[44, 58, 45, 57], X[19, 59, 20, 58],
> X[6, 32, 7, 31], X[45, 33, 46, 32], X[20, 34, 21, 33], X[59, 35, 60, 34],
> X[46, 8, 47, 7], X[21, 9, 22, 8], X[60, 10, 61, 9], X[35, 11, 36, 10],
> X[22, 48, 23, 47], X[61, 49, 62, 48], X[36, 50, 37, 49], X[11, 51, 12, 50],
> X[62, 24, 63, 23], X[37, 25, 38, 24], X[12, 26, 13, 25], X[51, 27, 52, 26],
> X[38, 64, 39, 63], X[13, 1, 14, 64], X[52, 2, 53, 1], X[27, 3, 28, 2],
> X[14, 40, 15, 39], X[53, 41, 54, 40], X[28, 42, 29, 41], X[3, 43, 4, 42]] |
In[5]:= | GaussCode[TorusKnot[8, 5]] |
Out[5]= | GaussCode[27, 28, -32, -3, -6, -9, 13, 14, 15, 16, -20, -23, -26, -29, 1, 2, 3,
> 4, -8, -11, -14, -17, 21, 22, 23, 24, -28, -31, -2, -5, 9, 10, 11, 12, -16,
> -19, -22, -25, 29, 30, 31, 32, -4, -7, -10, -13, 17, 18, 19, 20, -24, -27,
> -30, -1, 5, 6, 7, 8, -12, -15, -18, -21, 25, 26] |
In[6]:= | BR[TorusKnot[8, 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, 1, 2, 3, 4}] |
In[7]:= | alex = Alexander[TorusKnot[8, 5]][t] |
Out[7]= | -14 -13 -9 -8 -6 -5 -4 -3 1 3 4 5
-1 + t - t + t - t + t - t + t - t + - + t - t + t - t +
t
6 8 9 13 14
> t - t + t - t + t |
In[8]:= | Conway[TorusKnot[8, 5]][z] |
Out[8]= | 2 4 6 8 10 12
1 + 63 z + 1092 z + 8169 z + 32055 z + 73876 z + 107849 z +
14 16 18 20 22 24
> 104771 z + 69785 z + 32320 z + 10395 z + 2277 z + 324 z +
26 28
> 27 z + z |
In[9]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[9]= | {} |
In[10]:= | {KnotDet[TorusKnot[8, 5]], KnotSignature[TorusKnot[8, 5]]} |
Out[10]= | {5, 20} |
In[11]:= | J=Jones[TorusKnot[8, 5]][q] |
Out[11]= | 14 16 18 23 25
q + q + q - q - q |
In[12]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[12]= | {} |
In[13]:= | A2Invariant[TorusKnot[8, 5]][q] |
Out[13]= | NotAvailable |
In[14]:= | Kauffman[TorusKnot[8, 5]][a, z] |
Out[14]= | NotAvailable |
In[15]:= | {Vassiliev[2][TorusKnot[8, 5]], Vassiliev[3][TorusKnot[8, 5]]} |
Out[15]= | {63, 420} |
In[16]:= | Kh[TorusKnot[8, 5]][q, t] |
Out[16]= | 27 29 31 2 35 3 33 4 35 4 37 5 39 5 35 6
q + q + q t + q t + q t + q t + q t + q t + q t +
37 6 39 7 41 7 37 8 39 8 41 9 43 9
> q t + q t + q t + q t + 2 q t + q t + 2 q t +
41 10 45 11 43 12 45 12 49 12 47 13
> 2 q t + 3 q t + 2 q t + 2 q t + q t + 3 q t +
49 13 47 14 51 14 49 15 51 15 49 16 51 16
> 2 q t + 2 q t + q t + q t + 3 q t + q t + q t +
53 16 55 16 53 17 55 17 53 18 57 18 57 19
> q t + q t + 2 q t + q t + q t + q t + q t |