PD Presentation: |
X51,37,52,36 X66,38,67,37 X9,39,10,38 X24,40,25,39 X67,53,68,52 X10,54,11,53 X25,55,26,54 X40,56,41,55 X11,69,12,68 X26,70,27,69 X41,71,42,70 X56,72,57,71 X27,13,28,12 X42,14,43,13 X57,15,58,14 X72,16,1,15 X43,29,44,28 X58,30,59,29 X1,31,2,30 X16,32,17,31 X59,45,60,44 X2,46,3,45 X17,47,18,46 X32,48,33,47 X3,61,4,60 X18,62,19,61 X33,63,34,62 X48,64,49,63 X19,5,20,4 X34,6,35,5 X49,7,50,6 X64,8,65,7 X35,21,36,20 X50,22,51,21 X65,23,66,22 X8,24,9,23 |
Gauss Code: |
{-19, -22, -25, 29, 30, 31, 32, -36, -3, -6, -9, 13, 14, 15, 16, -20, -23, -26, -29, 33, 34, 35, 36, -4, -7, -10, -13, 17, 18, 19, 20, -24, -27, -30, -33, 1, 2, 3, 4, -8, -11, -14, -17, 21, 22, 23, 24, -28, -31, -34, -1, 5, 6, 7, 8, -12, -15, -18, -21, 25, 26, 27, 28, -32, -35, -2, -5, 9, 10, 11, 12, -16} |
In[1]:= |
<< KnotTheory` |
Loading KnotTheory` (version of August 30, 2005, 10:15:35)... |
In[2]:= | TubePlot[TorusKnot[9, 5]] |
|  |
Out[2]= | -Graphics- |
In[3]:= | Crossings[TorusKnot[9, 5]] |
Out[3]= | 36 |
In[4]:= | PD[TorusKnot[9, 5]] |
Out[4]= | PD[X[51, 37, 52, 36], X[66, 38, 67, 37], X[9, 39, 10, 38], X[24, 40, 25, 39],
> X[67, 53, 68, 52], X[10, 54, 11, 53], X[25, 55, 26, 54], X[40, 56, 41, 55],
> X[11, 69, 12, 68], X[26, 70, 27, 69], X[41, 71, 42, 70], X[56, 72, 57, 71],
> X[27, 13, 28, 12], X[42, 14, 43, 13], X[57, 15, 58, 14], X[72, 16, 1, 15],
> X[43, 29, 44, 28], X[58, 30, 59, 29], X[1, 31, 2, 30], X[16, 32, 17, 31],
> X[59, 45, 60, 44], X[2, 46, 3, 45], X[17, 47, 18, 46], X[32, 48, 33, 47],
> X[3, 61, 4, 60], X[18, 62, 19, 61], X[33, 63, 34, 62], X[48, 64, 49, 63],
> X[19, 5, 20, 4], X[34, 6, 35, 5], X[49, 7, 50, 6], X[64, 8, 65, 7],
> X[35, 21, 36, 20], X[50, 22, 51, 21], X[65, 23, 66, 22], X[8, 24, 9, 23]] |
In[5]:= | GaussCode[TorusKnot[9, 5]] |
Out[5]= | GaussCode[-19, -22, -25, 29, 30, 31, 32, -36, -3, -6, -9, 13, 14, 15, 16, -20,
> -23, -26, -29, 33, 34, 35, 36, -4, -7, -10, -13, 17, 18, 19, 20, -24, -27,
> -30, -33, 1, 2, 3, 4, -8, -11, -14, -17, 21, 22, 23, 24, -28, -31, -34, -1,
> 5, 6, 7, 8, -12, -15, -18, -21, 25, 26, 27, 28, -32, -35, -2, -5, 9, 10,
> 11, 12, -16] |
In[6]:= | BR[TorusKnot[9, 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, 1, 2, 3, 4}] |
In[7]:= | alex = Alexander[TorusKnot[9, 5]][t] |
Out[7]= | -16 -15 -11 -10 -7 -5 -2 2 5 7 10 11
-1 + t - t + t - t + t - t + t + t - t + t - t + t -
15 16
> t + t |
In[8]:= | Conway[TorusKnot[9, 5]][z] |
Out[8]= | 2 4 6 8 10 12
1 + 80 z + 1772 z + 17094 z + 87560 z + 267421 z + 526423 z +
14 16 18 20 22 24
> 703851 z + 661810 z + 447240 z + 219625 z + 78431 z + 20150 z +
26 28 30 32
> 3627 z + 434 z + 31 z + z |
In[9]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[9]= | {} |
In[10]:= | {KnotDet[TorusKnot[9, 5]], KnotSignature[TorusKnot[9, 5]]} |
Out[10]= | {1, 24} |
In[11]:= | J=Jones[TorusKnot[9, 5]][q] |
Out[11]= | 16 18 20 26 28
q + q + q - q - q |
In[12]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[12]= | {} |
In[13]:= | A2Invariant[TorusKnot[9, 5]][q] |
Out[13]= | NotAvailable |
In[14]:= | Kauffman[TorusKnot[9, 5]][a, z] |
Out[14]= | NotAvailable |
In[15]:= | {Vassiliev[2][TorusKnot[9, 5]], Vassiliev[3][TorusKnot[9, 5]]} |
Out[15]= | {80, 600} |
In[16]:= | Kh[TorusKnot[9, 5]][q, t] |
Out[16]= | 31 33 35 2 39 3 37 4 39 4 41 5 43 5 39 6
q + q + q t + q t + q t + q t + q t + q t + q t +
41 6 43 7 45 7 41 8 43 8 45 9 47 9
> q t + q t + q t + q t + 2 q t + q t + 2 q t +
45 10 49 11 47 12 49 12 53 12 51 13
> 2 q t + 3 q t + 2 q t + 2 q t + q t + 3 q t +
53 13 49 14 51 14 55 14 53 15 55 15
> 2 q t + q t + 2 q t + q t + 2 q t + 3 q t +
53 16 57 16 59 16 57 17 55 18 57 18 61 18
> 2 q t + q t + q t + 3 q t + q t + q t + q t +
59 19 61 19 59 20 63 20 63 21
> 2 q t + q t + q t + q t + q t |