© | Dror Bar-Natan: The Knot Atlas: Torus Knots:
T(4,3)
T(4,3) 		T(5,3)
T(5,3)
T(9,2)
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   The 9-Crossing Torus Knot T(9,2)

Visit T(9,2)'s page at Knotilus!

Acknowledgement

PD Presentation: X7,17,8,16 X17,9,18,8 X9,1,10,18 X1,11,2,10 X11,3,12,2 X3,13,4,12 X13,5,14,4 X5,15,6,14 X15,7,16,6

Gauss Code: {-4, 5, -6, 7, -8, 9, -1, 2, -3, 4, -5, 6, -7, 8, -9, 1, -2, 3}

Braid Representative:    

Alexander Polynomial: t-4 - t-3 + t-2 - t-1 + 1 - t + t2 - t3 + t4

Conway Polynomial: 1 + 10z2 + 15z4 + 7z6 + z8

Other knots with the same Alexander/Conway Polynomial: {91, ...}

Determinant and Signature: {9, 8}

Jones Polynomial: q4 + q6 - q7 + q8 - q9 + q10 - q11 + q12 - q13

Other knots (up to mirrors) with the same Jones Polynomial: {91, ...}

Coloured Jones Polynomial (in the 3-dimensional representation of sl(2); n=2): q8 + q11 - q13 + q14 - q16 + q17 - q19 + q20 - q22 + q23 - q25 + q26 - q27 - q28 + q29 - q31 + q32 - q34 + q35

A2 (sl(3)) Invariant: q14 + q16 + 2q18 + q20 + q22 - q34 - q36 - q38

Kauffman Polynomial: a-17z + a-16z2 - a-15z + a-15z3 - 2a-14z2 + a-14z4 + a-13z - 3a-13z3 + a-13z5 + 3a-12z2 - 4a-12z4 + a-12z6 - a-11z + 6a-11z3 - 5a-11z5 + a-11z7 + 4a-10 - 14a-10z2 + 16a-10z4 - 7a-10z6 + a-10z8 - 4a-9z + 10a-9z3 - 6a-9z5 + a-9z7 + 5a-8 - 20a-8z2 + 21a-8z4 - 8a-8z6 + a-8z8

V2 and V3, the type 2 and 3 Vassiliev invariants: {10, 30}

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(9,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 0123456789χ
27         1-1
25          0
23       11 0
21          0
19     11   0
17          0
15   11     0
13          0
11  1       1
91         1
71         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[9, 2]]
Out[2]=   
 -Graphics- 
In[3]:=
Crossings[TorusKnot[9, 2]]
Out[3]=   
9
In[4]:=
PD[TorusKnot[9, 2]]
Out[4]=   
PD[X[7, 17, 8, 16], X[17, 9, 18, 8], X[9, 1, 10, 18], X[1, 11, 2, 10], 
 
>   X[11, 3, 12, 2], X[3, 13, 4, 12], X[13, 5, 14, 4], X[5, 15, 6, 14], 
 
>   X[15, 7, 16, 6]]
In[5]:=
GaussCode[TorusKnot[9, 2]]
Out[5]=   
GaussCode[-4, 5, -6, 7, -8, 9, -1, 2, -3, 4, -5, 6, -7, 8, -9, 1, -2, 3]
In[6]:=
BR[TorusKnot[9, 2]]
Out[6]=   
BR[2, {1, 1, 1, 1, 1, 1, 1, 1, 1}]
In[7]:=
alex = Alexander[TorusKnot[9, 2]][t]
Out[7]=   
     -4    -3    -2   1        2    3    4
1 + t   - t   + t   - - - t + t  - t  + t
                      t
In[8]:=
Conway[TorusKnot[9, 2]][z]
Out[8]=   
        2       4      6    8
1 + 10 z  + 15 z  + 7 z  + z
In[9]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=   
{Knot[9, 1]}
In[10]:=
{KnotDet[TorusKnot[9, 2]], KnotSignature[TorusKnot[9, 2]]}
Out[10]=   
{9, 8}
In[11]:=
J=Jones[TorusKnot[9, 2]][q]
Out[11]=   
 4    6    7    8    9    10    11    12    13
q  + q  - q  + q  - q  + q   - q   + q   - q
In[12]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[12]=   
{Knot[9, 1]}
In[13]:=
ColouredJones[TorusKnot[9, 2], 2][q]
Out[13]=   
 8    11    13    14    16    17    19    20    22    23    25    26    27
q  + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   - 
 
     28    29    31    32    34    35
>   q   + q   - q   + q   - q   + q
In[14]:=
A2Invariant[TorusKnot[9, 2]][q]
Out[14]=   
 14    16      18    20    22    34    36    38
q   + q   + 2 q   + q   + q   - q   - q   - q
In[15]:=
Kauffman[TorusKnot[9, 2]][a, z]
Out[15]=   
                                          2       2      2       2       2
 4    5     z     z     z     z    4 z   z     2 z    3 z    14 z    20 z
--- + -- + --- - --- + --- - --- - --- + --- - ---- + ---- - ----- - ----- + 
 10    8    17    15    13    11    9     16    14     12      10      8
a     a    a     a     a     a     a     a     a      a       a       a
 
     3       3      3       3    4       4       4       4    5       5
    z     3 z    6 z    10 z    z     4 z    16 z    21 z    z     5 z
>   --- - ---- + ---- + ----- + --- - ---- + ----- + ----- + --- - ---- - 
     15    13     11      9      14    12      10      8      13    11
    a     a      a       a      a     a       a       a      a     a
 
       5    6       6      6    7     7    8     8
    6 z    z     7 z    8 z    z     z    z     z
>   ---- + --- - ---- - ---- + --- + -- + --- + --
      9     12    10      8     11    9    10    8
     a     a     a       a     a     a    a     a
In[16]:=
{Vassiliev[2][TorusKnot[9, 2]], Vassiliev[3][TorusKnot[9, 2]]}
Out[16]=   
{10, 30}
In[17]:=
Kh[TorusKnot[9, 2]][q, t]
Out[17]=   
 7    9    11  2    15  3    15  4    19  5    19  6    23  7    23  8    27  9
q  + q  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t

Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(9,2)
T(4,3)
T(4,3) 		T(5,3)
T(5,3)