© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(7,5)
T(7,5)
T(29,2)
T(29,2)
T(14,3)
TubePlot
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   The 28-Crossing Torus Knot T(14,3)

Visit T(14,3)'s page at Knotilus!

Acknowledgement

PD Presentation: X3,41,4,40 X22,42,23,41 X23,5,24,4 X42,6,43,5 X43,25,44,24 X6,26,7,25 X7,45,8,44 X26,46,27,45 X27,9,28,8 X46,10,47,9 X47,29,48,28 X10,30,11,29 X11,49,12,48 X30,50,31,49 X31,13,32,12 X50,14,51,13 X51,33,52,32 X14,34,15,33 X15,53,16,52 X34,54,35,53 X35,17,36,16 X54,18,55,17 X55,37,56,36 X18,38,19,37 X19,1,20,56 X38,2,39,1 X39,21,40,20 X2,22,3,21

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

Braid Representative:    

Alexander Polynomial: t-13 - t-12 + t-10 - t-9 + t-7 - t-6 + t-4 - t-3 + t-1 - 1 + t - t3 + t4 - t6 + t7 - t9 + t10 - t12 + t13

Conway Polynomial: 1 + 65z2 + 1040z4 + 6448z6 + 20540z8 + 38532z10 + 45942z12 + 36366z14 + 19532z16 + 7144z18 + 1751z20 + 275z22 + 25z24 + z26

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

Determinant and Signature: {3, 18}

Jones Polynomial: q13 + q15 - q28

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: {65, 455}

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(14,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
012345678910111213141516171819χ
57                   1-1
55                 1  -1
53                 11 0
51               11   0
49             1  1   0
47             11     0
45           11       0
43         1  1       0
41         11         0
39       11           0
37     1  1           0
35     11             0
33   11               0
31    1               1
29  1                 1
271                   1
251                   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[14, 3]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[14, 3]]
Out[3]=   
28
In[4]:=
PD[TorusKnot[14, 3]]
Out[4]=   
PD[X[3, 41, 4, 40], X[22, 42, 23, 41], X[23, 5, 24, 4], X[42, 6, 43, 5], 
 
>   X[43, 25, 44, 24], X[6, 26, 7, 25], X[7, 45, 8, 44], X[26, 46, 27, 45], 
 
>   X[27, 9, 28, 8], X[46, 10, 47, 9], X[47, 29, 48, 28], X[10, 30, 11, 29], 
 
>   X[11, 49, 12, 48], X[30, 50, 31, 49], X[31, 13, 32, 12], X[50, 14, 51, 13], 
 
>   X[51, 33, 52, 32], X[14, 34, 15, 33], X[15, 53, 16, 52], X[34, 54, 35, 53], 
 
>   X[35, 17, 36, 16], X[54, 18, 55, 17], X[55, 37, 56, 36], X[18, 38, 19, 37], 
 
>   X[19, 1, 20, 56], X[38, 2, 39, 1], X[39, 21, 40, 20], X[2, 22, 3, 21]]
In[5]:=
GaussCode[TorusKnot[14, 3]]
Out[5]=   
GaussCode[26, -28, -1, 3, 4, -6, -7, 9, 10, -12, -13, 15, 16, -18, -19, 21, 22, 
 
>   -24, -25, 27, 28, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 17, 18, -20, -21, 
 
>   23, 24, -26, -27, 1, 2, -4, -5, 7, 8, -10, -11, 13, 14, -16, -17, 19, 20, 
 
>   -22, -23, 25]
In[6]:=
BR[TorusKnot[14, 3]]
Out[6]=   
BR[3, {1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 
 
>    1, 2, 1, 2}]
In[7]:=
alex = Alexander[TorusKnot[14, 3]][t]
Out[7]=   
      -13    -12    -10    -9    -7    -6    -4    -3   1        3    4    6
-1 + t    - t    + t    - t   + t   - t   + t   - t   + - + t - t  + t  - t  + 
                                                        t
 
     7    9    10    12    13
>   t  - t  + t   - t   + t
In[8]:=
Conway[TorusKnot[14, 3]][z]
Out[8]=   
        2         4         6          8          10          12          14
1 + 65 z  + 1040 z  + 6448 z  + 20540 z  + 38532 z   + 45942 z   + 36366 z   + 
 
           16         18         20        22       24    26
>   19532 z   + 7144 z   + 1751 z   + 275 z   + 25 z   + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{}
In[10]:=
{KnotDet[TorusKnot[14, 3]], KnotSignature[TorusKnot[14, 3]]}
Out[10]=   
{3, 18}
In[11]:=
J=Jones[TorusKnot[14, 3]][q]
Out[11]=   
 13    15    28
q   + q   - q
In[12]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[12]=   
{}
In[13]:=
A2Invariant[TorusKnot[14, 3]][q]
Out[13]=   
NotAvailable
In[14]:=
Kauffman[TorusKnot[14, 3]][a, z]
Out[14]=   
NotAvailable
In[15]:=
{Vassiliev[2][TorusKnot[14, 3]], Vassiliev[3][TorusKnot[14, 3]]}
Out[15]=   
{65, 455}
In[16]:=
Kh[TorusKnot[14, 3]][q, t]
Out[16]=   
 25    27    29  2    33  3    31  4    33  4    35  5    37  5    35  6
q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 
 
     39  7    37  8    39  8    41  9    43  9    41  10    45  11    43  12
>   q   t  + q   t  + q   t  + q   t  + q   t  + q   t   + q   t   + q   t   + 
 
     45  12    47  13    49  13    47  14    51  15    49  16    51  16
>   q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + 
 
     53  17    55  17    53  18    57  19
>   q   t   + q   t   + q   t   + q   t


Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(14,3)
T(7,5)
T(7,5)
T(29,2)
T(29,2)