T(7,2)

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[[Image:T(5,2).{{{ext}}}|80px|link=T(5,2)]]

T(5,2)

[[Image:T(4,3).{{{ext}}}|80px|link=T(4,3)]]

T(4,3)

Visit T(7,2)'s page at Knotilus!

Visit T(7,2)'s page at the original Knot Atlas!

Knot presentations

Planar diagram 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}
Dowker-Thistlethwaite code 8 10 12 14 2 4 6

Polynomial invariants

Polynomial invariants

Alexander polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^3-t^2+t-1+ t^{-1} - t^{-2} + t^{-3} }
Conway polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6+5 z^4+6 z^2+1}
2nd Alexander ideal (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature { 7, 6 }
Jones polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{10}+q^9-q^8+q^7-q^6+q^5+q^3}
HOMFLY-PT polynomial (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6 a^{-6} +6 z^4 a^{-6} -z^4 a^{-8} +10 z^2 a^{-6} -4 z^2 a^{-8} +4 a^{-6} -3 a^{-8} }
Kauffman polynomial (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6 a^{-6} +z^6 a^{-8} +z^5 a^{-7} +z^5 a^{-9} -6 z^4 a^{-6} -5 z^4 a^{-8} +z^4 a^{-10} -4 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} +10 z^2 a^{-6} +7 z^2 a^{-8} -2 z^2 a^{-10} +z^2 a^{-12} +3 z a^{-7} +z a^{-9} -z a^{-11} +z a^{-13} -4 a^{-6} -3 a^{-8} }
The A2 invariant Data:T(7,2)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(7,2)/QuantumInvariant/G2/1,0
Further Quantum Invariants
T(7,2) Quantum Invariants.
Computer Talk
The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["T(7,2)"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^3-t^2+t-1+ t^{-1} - t^{-2} + t^{-3} }
In[5]:= Conway[K][z]
Out[5]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6+5 z^4+6 z^2+1}
In[6]:= Alexander[K, 2][t]
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
Out[6]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 7, 6 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{10}+q^9-q^8+q^7-q^6+q^5+q^3}
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6 a^{-6} +6 z^4 a^{-6} -z^4 a^{-8} +10 z^2 a^{-6} -4 z^2 a^{-8} +4 a^{-6} -3 a^{-8} }
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6 a^{-6} +z^6 a^{-8} +z^5 a^{-7} +z^5 a^{-9} -6 z^4 a^{-6} -5 z^4 a^{-8} +z^4 a^{-10} -4 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} +10 z^2 a^{-6} +7 z^2 a^{-8} -2 z^2 a^{-10} +z^2 a^{-12} +3 z a^{-7} +z a^{-9} -z a^{-11} +z a^{-13} -4 a^{-6} -3 a^{-8} }

Vassiliev invariants

V2 and V3 {0, 14})

Khovanov Homology. The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Include[ColouredJonesM.mhtml] 

In[1]:=    
 
<< KnotTheory`
 
Loading KnotTheory` (version of August 19, 2005, 13:11:25)...
In[2]:=
Crossings[TorusKnot[7, 2]]
Out[2]=   
7
In[3]:=
PD[TorusKnot[7, 2]]
Out[3]=   
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[4]:=
GaussCode[TorusKnot[7, 2]]
Out[4]=   
GaussCode[-4, 5, -6, 7, -1, 2, -3, 4, -5, 6, -7, 1, -2, 3]
In[5]:=
BR[TorusKnot[7, 2]]
Out[5]=   
BR[2, {1, 1, 1, 1, 1, 1, 1}]
In[6]:=
alex = Alexander[TorusKnot[7, 2]][t]
Out[6]=   
      -3    -2   1        2    3

-1 + t   - t   + - + t - t  + t


                 t
In[7]:=
Conway[TorusKnot[7, 2]][z]
Out[7]=   
       2      4    6
1 + 6 z  + 5 z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[7, 1]}
In[9]:=
{KnotDet[TorusKnot[7, 2]], KnotSignature[TorusKnot[7, 2]]}
Out[9]=   
{7, 6}
In[10]:=
J=Jones[TorusKnot[7, 2]][q]
Out[10]=   
 3    5    6    7    8    9    10
q  + q  - q  + q  - q  + q  - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[7, 1]}
In[12]:=
A2Invariant[TorusKnot[7, 2]][q]
Out[12]=   
 10    12      14    16    18    26    28    30
q   + q   + 2 q   + q   + q   - q   - q   - q
In[13]:=
Kauffman[TorusKnot[7, 2]][a, z]
Out[13]=   
                                  2       2      2       2    3

-3   4     z     z    z    3 z   z     2 z    7 z    10 z    z
-- - -- + --- - --- + -- + --- + --- - ---- + ---- + ----- + --- - 



8    6    13    11    9    7     12    10      8      6      11



a    a    a     a     a    a     a     a       a      a      a



    3      3    4       4      4    5    5    6    6
 3 z    4 z    z     5 z    6 z    z    z    z    z
 ---- - ---- + --- - ---- - ---- + -- + -- + -- + --
   9      7     10     8      6     9    7    8    6


   a      a     a      a      a     a    a    a    a
In[14]:=
{Vassiliev[2][TorusKnot[7, 2]], Vassiliev[3][TorusKnot[7, 2]]}
Out[14]=   
{0, 14}
In[15]:=
Kh[TorusKnot[7, 2]][q, t]
Out[15]=   
 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
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