© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(13,2)
T(13,2) 		T(5,4)
T(5,4)
T(7,3)
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   The 14-Crossing Torus Knot T(7,3)

Visit T(7,3)'s page at Knotilus!

Acknowledgement

PD Presentation: X1,11,2,10 X20,12,21,11 X21,3,22,2 X12,4,13,3 X13,23,14,22 X4,24,5,23 X5,15,6,14 X24,16,25,15 X25,7,26,6 X16,8,17,7 X17,27,18,26 X8,28,9,27 X9,19,10,18 X28,20,1,19

Gauss Code: {-1, 3, 4, -6, -7, 9, 10, -12, -13, 1, 2, -4, -5, 7, 8, -10, -11, 13, 14, -2, -3, 5, 6, -8, -9, 11, 12, -14}

Braid Representative:    

Alexander Polynomial: t-6 - t-5 + t-3 - t-2 + 1 - t2 + t3 - t5 + t6

Conway Polynomial: 1 + 16z2 + 60z4 + 78z6 + 44z8 + 11z10 + z12

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

Determinant and Signature: {1, 8}

Jones Polynomial: q6 + q8 - q14

Other knots (up to mirrors) with the same Jones Polynomial: {...}

Coloured Jones Polynomial (in the 3-dimensional representation of sl(2); n=2): q12 + q15 + q18 - q19 + q21 - q22 + q24 - q25 - q28 - q31 + q32 - q34 + q35 - q37 + q38 - q40 + q41

A2 (sl(3)) Invariant: q22 + q24 + 2q26 + 2q28 + 2q30 + q32 + q34 - q38 - 2q40 - 2q42 - 2q44 - q46 + q56

Kauffman Polynomial: 5a-16 - 10a-16z2 + 6a-16z4 - a-16z6 - 16a-15z + 60a-15z3 - 78a-15z5 + 44a-15z7 - 11a-15z9 + a-15z11 + 16a-14 - 76a-14z2 + 138a-14z4 - 122a-14z6 + 55a-14z8 - 12a-14z10 + a-14z12 - 16a-13z + 60a-13z3 - 78a-13z5 + 44a-13z7 - 11a-13z9 + a-13z11 + 12a-12 - 66a-12z2 + 132a-12z4 - 121a-12z6 + 55a-12z8 - 12a-12z10 + a-12z12

V2 and V3, the type 2 and 3 Vassiliev invariants: {16, 56}

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=8 is the signature of T(7,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 0123456789χ
29         1-1
27         1-1
25       11 0
23     1  1 0
21     11   0
19   11     0
17    1     1
15  1       1
131         1
111         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, 3]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[7, 3]]
Out[3]=   
14
In[4]:=
PD[TorusKnot[7, 3]]
Out[4]=   
PD[X[1, 11, 2, 10], X[20, 12, 21, 11], X[21, 3, 22, 2], X[12, 4, 13, 3], 
 
>   X[13, 23, 14, 22], X[4, 24, 5, 23], X[5, 15, 6, 14], X[24, 16, 25, 15], 
 
>   X[25, 7, 26, 6], X[16, 8, 17, 7], X[17, 27, 18, 26], X[8, 28, 9, 27], 
 
>   X[9, 19, 10, 18], X[28, 20, 1, 19]]
In[5]:=
GaussCode[TorusKnot[7, 3]]
Out[5]=   
GaussCode[-1, 3, 4, -6, -7, 9, 10, -12, -13, 1, 2, -4, -5, 7, 8, -10, -11, 13, 
 
>   14, -2, -3, 5, 6, -8, -9, 11, 12, -14]
In[6]:=
BR[TorusKnot[7, 3]]
Out[6]=   
BR[3, {1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2}]
In[7]:=
alex = Alexander[TorusKnot[7, 3]][t]
Out[7]=   
     -6    -5    -3    -2    2    3    5    6
1 + t   - t   + t   - t   - t  + t  - t  + t
In[8]:=
Conway[TorusKnot[7, 3]][z]
Out[8]=   
        2       4       6       8       10    12
1 + 16 z  + 60 z  + 78 z  + 44 z  + 11 z   + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{}
In[10]:=
{KnotDet[TorusKnot[7, 3]], KnotSignature[TorusKnot[7, 3]]}
Out[10]=   
{1, 8}
In[11]:=
J=Jones[TorusKnot[7, 3]][q]
Out[11]=   
 6    8    14
q  + q  - q
In[12]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[12]=   
{}
In[13]:=
ColouredJones[TorusKnot[7, 3], 2][q]
Out[13]=   
 12    15    18    19    21    22    24    25    28    31    32    34    35
q   + q   + q   - q   + q   - q   + q   - q   - q   - q   + q   - q   + q   - 
 
     37    38    40    41
>   q   + q   - q   + q
In[14]:=
A2Invariant[TorusKnot[7, 3]][q]
Out[14]=   
 22    24      26      28      30    32    34    38      40      42      44
q   + q   + 2 q   + 2 q   + 2 q   + q   + q   - q   - 2 q   - 2 q   - 2 q   - 
 
     46    56
>   q   + q
In[15]:=
Kauffman[TorusKnot[7, 3]][a, z]
Out[15]=   
                                    2       2       2       3       3      4
 5    16    12    16 z   16 z   10 z    76 z    66 z    60 z    60 z    6 z
--- + --- + --- - ---- - ---- - ----- - ----- - ----- + ----- + ----- + ---- + 
 16    14    12    15     13      16      14      12      15      13     16
a     a     a     a      a       a       a       a       a       a      a
 
         4        4       5       5    6         6        6       7       7
    138 z    132 z    78 z    78 z    z     122 z    121 z    44 z    44 z
>   ------ + ------ - ----- - ----- - --- - ------ - ------ + ----- + ----- + 
      14       12       15      13     16     14       12       15      13
     a        a        a       a      a      a        a        a       a
 
        8       8       9       9       10       10    11    11    12    12
    55 z    55 z    11 z    11 z    12 z     12 z     z     z     z     z
>   ----- + ----- - ----- - ----- - ------ - ------ + --- + --- + --- + ---
      14      12      15      13      14       12      15    13    14    12
     a       a       a       a       a        a       a     a     a     a
In[16]:=
{Vassiliev[2][TorusKnot[7, 3]], Vassiliev[3][TorusKnot[7, 3]]}
Out[16]=   
{16, 56}
In[17]:=
Kh[TorusKnot[7, 3]][q, t]
Out[17]=   
 11    13    15  2    19  3    17  4    19  4    21  5    23  5    21  6
q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 
 
     25  7    23  8    25  8    27  9    29  9
>   q   t  + q   t  + q   t  + q   t  + q   t

Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(7,3)
T(13,2)
T(13,2) 		T(5,4)
T(5,4)