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

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Acknowledgement

PD Presentation: X5,13,6,12 X13,7,14,6 X7,1,8,14 X1928 X9,3,10,2 X3,11,4,10 X11,5,12,4

Gauss Code: {-4, 5, -6, 7, -1, 2, -3, 4, -5, 6, -7, 1, -2, 3}

Braid Representative:    

Alexander Polynomial: t-3 - t-2 + t-1 - 1 + t - t2 + t3

Conway Polynomial: 1 + 6z2 + 5z4 + z6

Other knots with the same Alexander/Conway Polynomial: {71, ...}

Determinant and Signature: {7, 6}

Jones Polynomial: q3 + q5 - q6 + q7 - q8 + q9 - q10

Other knots (up to mirrors) with the same Jones Polynomial: {71, ...}

Coloured Jones Polynomial (in the 3-dimensional representation of sl(2); n=2): q6 + q9 - q11 + q12 - q14 + q15 - q17 + q18 - q20 - q23 + q24 - q26 + q27

A2 (sl(3)) Invariant: q10 + q12 + 2q14 + q16 + q18 - q26 - q28 - q30

Kauffman Polynomial: a-13z + a-12z2 - a-11z + a-11z3 - 2a-10z2 + a-10z4 + a-9z - 3a-9z3 + a-9z5 - 3a-8 + 7a-8z2 - 5a-8z4 + a-8z6 + 3a-7z - 4a-7z3 + a-7z5 - 4a-6 + 10a-6z2 - 6a-6z4 + a-6z6

V2 and V3, the type 2 and 3 Vassiliev invariants: {6, 14}

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=6 is the signature of T(7,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
01234567χ
21       1-1
19        0
17     11 0
15        0
13   11   0
11        0
9  1     1
71       1
51       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, 2]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[7, 2]]
Out[3]=   
7
In[4]:=
PD[TorusKnot[7, 2]]
Out[4]=   
PD[X[5, 13, 6, 12], X[13, 7, 14, 6], X[7, 1, 8, 14], X[1, 9, 2, 8], 
 
>   X[9, 3, 10, 2], X[3, 11, 4, 10], X[11, 5, 12, 4]]
In[5]:=
GaussCode[TorusKnot[7, 2]]
Out[5]=   
GaussCode[-4, 5, -6, 7, -1, 2, -3, 4, -5, 6, -7, 1, -2, 3]
In[6]:=
BR[TorusKnot[7, 2]]
Out[6]=   
BR[2, {1, 1, 1, 1, 1, 1, 1}]
In[7]:=
alex = Alexander[TorusKnot[7, 2]][t]
Out[7]=   
      -3    -2   1        2    3
-1 + t   - t   + - + t - t  + t
                 t
In[8]:=
Conway[TorusKnot[7, 2]][z]
Out[8]=   
       2      4    6
1 + 6 z  + 5 z  + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{Knot[7, 1]}
In[10]:=
{KnotDet[TorusKnot[7, 2]], KnotSignature[TorusKnot[7, 2]]}
Out[10]=   
{7, 6}
In[11]:=
J=Jones[TorusKnot[7, 2]][q]
Out[11]=   
 3    5    6    7    8    9    10
q  + q  - q  + q  - q  + q  - q
In[12]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[12]=   
{Knot[7, 1]}
In[13]:=
ColouredJones[TorusKnot[7, 2], 2][q]
Out[13]=   
 6    9    11    12    14    15    17    18    20    23    24    26    27
q  + q  - q   + q   - q   + q   - q   + q   - q   - q   + q   - q   + q
In[14]:=
A2Invariant[TorusKnot[7, 2]][q]
Out[14]=   
 10    12      14    16    18    26    28    30
q   + q   + 2 q   + q   + q   - q   - q   - q
In[15]:=
Kauffman[TorusKnot[7, 2]][a, z]
Out[15]=   
                                  2       2      2       2    3       3
-3   4     z     z    z    3 z   z     2 z    7 z    10 z    z     3 z
-- - -- + --- - --- + -- + --- + --- - ---- + ---- + ----- + --- - ---- - 
 8    6    13    11    9    7     12    10      8      6      11     9
a    a    a     a     a    a     a     a       a      a      a      a
 
       3    4       4      4    5    5    6    6
    4 z    z     5 z    6 z    z    z    z    z
>   ---- + --- - ---- - ---- + -- + -- + -- + --
      7     10     8      6     9    7    8    6
     a     a      a      a     a    a    a    a
In[16]:=
{Vassiliev[2][TorusKnot[7, 2]], Vassiliev[3][TorusKnot[7, 2]]}
Out[16]=   
{6, 14}
In[17]:=
Kh[TorusKnot[7, 2]][q, t]
Out[17]=   
 5    7    9  2    13  3    13  4    17  5    17  6    21  7
q  + q  + q  t  + q   t  + q   t  + q   t  + q   t  + q   t


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