© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(9,5)
T(9,5) 		T(5,2)
T(5,2)
T(3,2)
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   The 3-Crossing Torus Knot T(3,2)

Visit T(3,2)'s page at Knotilus!

Acknowledgement

PD Presentation: X3146 X1524 X5362

Gauss Code: {-2, 3, -1, 2, -3, 1}

Braid Representative:    

Alexander Polynomial: t-1 - 1 + t

Conway Polynomial: 1 + z2

Other knots with the same Alexander/Conway Polynomial: {31, ...}

Determinant and Signature: {3, 2}

Jones Polynomial: q + q3 - q4

Other knots (up to mirrors) with the same Jones Polynomial: {31, ...}

Coloured Jones Polynomial (in the 3-dimensional representation of sl(2); n=2): q2 + q5 - q7 + q8 - q9 - q10 + q11

A2 (sl(3)) Invariant: q2 + q4 + 2q6 + q8 - q12 - q14

Kauffman Polynomial: a-5z - a-4 + a-4z2 + a-3z - 2a-2 + a-2z2

V2 and V3, the type 2 and 3 Vassiliev invariants: {1, 1}

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=2 is the signature of T(3,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 0123χ
9   1-1
7    0
5  1 1
31   1
11   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[3, 2]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[3, 2]]
Out[3]=   
3
In[4]:=
PD[TorusKnot[3, 2]]
Out[4]=   
PD[X[3, 1, 4, 6], X[1, 5, 2, 4], X[5, 3, 6, 2]]
In[5]:=
GaussCode[TorusKnot[3, 2]]
Out[5]=   
GaussCode[-2, 3, -1, 2, -3, 1]
In[6]:=
BR[TorusKnot[3, 2]]
Out[6]=   
BR[2, {1, 1, 1}]
In[7]:=
alex = Alexander[TorusKnot[3, 2]][t]
Out[7]=   
     1
-1 + - + t
     t
In[8]:=
Conway[TorusKnot[3, 2]][z]
Out[8]=   
     2
1 + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{Knot[3, 1]}
In[10]:=
{KnotDet[TorusKnot[3, 2]], KnotSignature[TorusKnot[3, 2]]}
Out[10]=   
{3, 2}
In[11]:=
J=Jones[TorusKnot[3, 2]][q]
Out[11]=   
     3    4
q + q  - q
In[12]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[12]=   
{Knot[3, 1]}
In[13]:=
ColouredJones[TorusKnot[3, 2], 2][q]
Out[13]=   
 2    5    7    8    9    10    11
q  + q  - q  + q  - q  - q   + q
In[14]:=
A2Invariant[TorusKnot[3, 2]][q]
Out[14]=   
 2    4      6    8    12    14
q  + q  + 2 q  + q  - q   - q
In[15]:=
Kauffman[TorusKnot[3, 2]][a, z]
Out[15]=   
                       2    2
  -4   2    z    z    z    z
-a   - -- + -- + -- + -- + --
        2    5    3    4    2
       a    a    a    a    a
In[16]:=
{Vassiliev[2][TorusKnot[3, 2]], Vassiliev[3][TorusKnot[3, 2]]}
Out[16]=   
{1, 1}
In[17]:=
Kh[TorusKnot[3, 2]][q, t]
Out[17]=   
     3    5  2    9  3
q + q  + q  t  + q  t

Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(3,2)
T(9,5)
T(9,5) 		T(5,2)
T(5,2)