PD Presentation: |
X17,49,18,48 X49,19,50,18 X19,51,20,50 X51,21,52,20 X21,53,22,52 X53,23,54,22 X23,55,24,54 X55,25,56,24 X25,57,26,56 X57,27,58,26 X27,59,28,58 X59,29,60,28 X29,61,30,60 X61,31,62,30 X31,1,32,62 X1,33,2,32 X33,3,34,2 X3,35,4,34 X35,5,36,4 X5,37,6,36 X37,7,38,6 X7,39,8,38 X39,9,40,8 X9,41,10,40 X41,11,42,10 X11,43,12,42 X43,13,44,12 X13,45,14,44 X45,15,46,14 X15,47,16,46 X47,17,48,16 |
Gauss Code: |
{-16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -28, 29, -30, 31, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15} |
In[1]:= |
<< KnotTheory` |
Loading KnotTheory` (version of August 30, 2005, 10:15:35)... |
In[2]:= | TubePlot[TorusKnot[31, 2]] |
|  |
Out[2]= | -Graphics- |
In[3]:= | Crossings[TorusKnot[31, 2]] |
Out[3]= | 31 |
In[4]:= | PD[TorusKnot[31, 2]] |
Out[4]= | PD[X[17, 49, 18, 48], X[49, 19, 50, 18], X[19, 51, 20, 50], X[51, 21, 52, 20],
> X[21, 53, 22, 52], X[53, 23, 54, 22], X[23, 55, 24, 54], X[55, 25, 56, 24],
> X[25, 57, 26, 56], X[57, 27, 58, 26], X[27, 59, 28, 58], X[59, 29, 60, 28],
> X[29, 61, 30, 60], X[61, 31, 62, 30], X[31, 1, 32, 62], X[1, 33, 2, 32],
> X[33, 3, 34, 2], X[3, 35, 4, 34], X[35, 5, 36, 4], X[5, 37, 6, 36],
> X[37, 7, 38, 6], X[7, 39, 8, 38], X[39, 9, 40, 8], X[9, 41, 10, 40],
> X[41, 11, 42, 10], X[11, 43, 12, 42], X[43, 13, 44, 12], X[13, 45, 14, 44],
> X[45, 15, 46, 14], X[15, 47, 16, 46], X[47, 17, 48, 16]] |
In[5]:= | GaussCode[TorusKnot[31, 2]] |
Out[5]= | GaussCode[-16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -28, 29, -30,
> 31, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18,
> -19, 20, -21, 22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 1, -2, 3, -4, 5,
> -6, 7, -8, 9, -10, 11, -12, 13, -14, 15] |
In[6]:= | BR[TorusKnot[31, 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, 1,
> 1, 1, 1, 1, 1, 1, 1}] |
In[7]:= | alex = Alexander[TorusKnot[31, 2]][t] |
Out[7]= | -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5
-1 + t - t + t - t + t - t + t - t + t - t + t -
-4 -3 -2 1 2 3 4 5 6 7 8 9 10
> t + t - t + - + t - t + t - t + t - t + t - t + t - t +
t
11 12 13 14 15
> t - t + t - t + t |
In[8]:= | Conway[TorusKnot[31, 2]][z] |
Out[8]= | 2 4 6 8 10 12
1 + 120 z + 2380 z + 18564 z + 75582 z + 184756 z + 293930 z +
14 16 18 20 22 24
> 319770 z + 245157 z + 134596 z + 53130 z + 14950 z + 2925 z +
26 28 30
> 378 z + 29 z + z |
In[9]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[9]= | {} |
In[10]:= | {KnotDet[TorusKnot[31, 2]], KnotSignature[TorusKnot[31, 2]]} |
Out[10]= | {31, 30} |
In[11]:= | J=Jones[TorusKnot[31, 2]][q] |
Out[11]= | 15 17 18 19 20 21 22 23 24 25 26 27 28
q + q - q + q - q + q - q + q - q + q - q + q - q +
29 30 31 32 33 34 35 36 37 38 39 40
> q - q + q - q + q - q + q - q + q - q + q - q +
41 42 43 44 45 46
> 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[31, 2]][q] |
Out[13]= | NotAvailable |
In[14]:= | Kauffman[TorusKnot[31, 2]][a, z] |
Out[14]= | NotAvailable |
In[15]:= | {Vassiliev[2][TorusKnot[31, 2]], Vassiliev[3][TorusKnot[31, 2]]} |
Out[15]= | {120, 1240} |
In[16]:= | Kh[TorusKnot[31, 2]][q, t] |
Out[16]= | 29 31 33 2 37 3 37 4 41 5 41 6 45 7 45 8
q + q + q t + q t + q t + q t + q t + q t + q t +
49 9 49 10 53 11 53 12 57 13 57 14 61 15
> q t + q t + q t + q t + q t + q t + q t +
61 16 65 17 65 18 69 19 69 20 73 21 73 22
> q t + q t + q t + q t + q t + q t + q t +
77 23 77 24 81 25 81 26 85 27 85 28 89 29
> q t + q t + q t + q t + q t + q t + q t +
89 30 93 31
> q t + q t |