T(11,3)

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[[Image:T(21,2).{{{ext}}}|80px|link=T(21,2)]]

T(21,2)

[[Image:T(23,2).{{{ext}}}|80px|link=T(23,2)]]

T(23,2)

T(11,3).jpg Visit T(11,3)'s page at Knotilus!

Visit T(11,3)'s page at the original Knot Atlas!

T(11,3) Quick Notes


T(11,3) Further Notes and Views

Knot presentations

Planar diagram presentation X7,37,8,36 X22,38,23,37 X23,9,24,8 X38,10,39,9 X39,25,40,24 X10,26,11,25 X11,41,12,40 X26,42,27,41 X27,13,28,12 X42,14,43,13 X43,29,44,28 X14,30,15,29 X15,1,16,44 X30,2,31,1 X31,17,32,16 X2,18,3,17 X3,33,4,32 X18,34,19,33 X19,5,20,4 X34,6,35,5 X35,21,36,20 X6,22,7,21
Gauss code {14, -16, -17, 19, 20, -22, -1, 3, 4, -6, -7, 9, 10, -12, -13, 15, 16, -18, -19, 21, 22, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 17, 18, -20, -21, 1, 2, -4, -5, 7, 8, -10, -11, 13}
Dowker-Thistlethwaite code 30 -32 34 -36 38 -40 42 -44 2 -4 6 -8 10 -12 14 -16 18 -20 22 -24 26 -28

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^{10}-t^9+t^7-t^6+t^4-t^3+t-1+ t^{-1} - t^{-3} + t^{-4} - t^{-6} + t^{-7} - t^{-9} + t^{-10} }
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^{20}+19 z^{18}+152 z^{16}+666 z^{14}+1742 z^{12}+2782 z^{10}+2665 z^8+1443 z^6+390 z^4+40 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 { 1, 16 }
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^{22}+q^{12}+q^{10}}
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^{20} a^{-20} +20 z^{18} a^{-20} -z^{18} a^{-22} +171 z^{16} a^{-20} -19 z^{16} a^{-22} +817 z^{14} a^{-20} -152 z^{14} a^{-22} +z^{14} a^{-24} +2394 z^{12} a^{-20} -666 z^{12} a^{-22} +14 z^{12} a^{-24} +4446 z^{10} a^{-20} -1742 z^{10} a^{-22} +78 z^{10} a^{-24} +5226 z^8 a^{-20} -2782 z^8 a^{-22} +221 z^8 a^{-24} +3770 z^6 a^{-20} -2665 z^6 a^{-22} +338 z^6 a^{-24} +1560 z^4 a^{-20} -1443 z^4 a^{-22} +273 z^4 a^{-24} +325 z^2 a^{-20} -390 z^2 a^{-22} +105 z^2 a^{-24} +26 a^{-20} -40 a^{-22} +15 a^{-24} }
Kauffman polynomial (db, data sources) Data:T(11,3)/Kauffman Polynomial
The A2 invariant Data:T(11,3)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(11,3)/QuantumInvariant/G2/1,0
Further Quantum Invariants
T(11,3) 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(11,3)"];
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^{10}-t^9+t^7-t^6+t^4-t^3+t-1+ t^{-1} - t^{-3} + t^{-4} - t^{-6} + t^{-7} - t^{-9} + t^{-10} }
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^{20}+19 z^{18}+152 z^{16}+666 z^{14}+1742 z^{12}+2782 z^{10}+2665 z^8+1443 z^6+390 z^4+40 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]= { 1, 16 }
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^{22}+q^{12}+q^{10}}
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^{20} a^{-20} +20 z^{18} a^{-20} -z^{18} a^{-22} +171 z^{16} a^{-20} -19 z^{16} a^{-22} +817 z^{14} a^{-20} -152 z^{14} a^{-22} +z^{14} a^{-24} +2394 z^{12} a^{-20} -666 z^{12} a^{-22} +14 z^{12} a^{-24} +4446 z^{10} a^{-20} -1742 z^{10} a^{-22} +78 z^{10} a^{-24} +5226 z^8 a^{-20} -2782 z^8 a^{-22} +221 z^8 a^{-24} +3770 z^6 a^{-20} -2665 z^6 a^{-22} +338 z^6 a^{-24} +1560 z^4 a^{-20} -1443 z^4 a^{-22} +273 z^4 a^{-24} +325 z^2 a^{-20} -390 z^2 a^{-22} +105 z^2 a^{-24} +26 a^{-20} -40 a^{-22} +15 a^{-24} }
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= Data:T(11,3)/Kauffman Polynomial

Vassiliev invariants

V2 and V3 {0, 220}

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=} 16 is the signature of T(11,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 0123456789101112131415χ
45               1-1
43             1  -1
41             11 0
39           11   0
37         1  1   0
35         11     0
33       11       0
31     1  1       0
29     11         0
27   11           0
25    1           1
23  1             1
211               1
191               1

Computer Talk

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

In[1]:=    
 
<< KnotTheory`
 
Loading KnotTheory` (version of August 19, 2005, 13:11:25)...
In[2]:=
Crossings[TorusKnot[11, 3]]
Out[2]=  
22
In[3]:=
PD[TorusKnot[11, 3]]
Out[3]=  
PD[X[7, 37, 8, 36], X[22, 38, 23, 37], X[23, 9, 24, 8], 

 X[38, 10, 39, 9], X[39, 25, 40, 24], X[10, 26, 11, 25], 

 X[11, 41, 12, 40], X[26, 42, 27, 41], X[27, 13, 28, 12], 

 X[42, 14, 43, 13], X[43, 29, 44, 28], X[14, 30, 15, 29], 

 X[15, 1, 16, 44], X[30, 2, 31, 1], X[31, 17, 32, 16], 

 X[2, 18, 3, 17], X[3, 33, 4, 32], X[18, 34, 19, 33], X[19, 5, 20, 4], 



  X[34, 6, 35, 5], X[35, 21, 36, 20], X[6, 22, 7, 21]]
In[4]:=
GaussCode[TorusKnot[11, 3]]
Out[4]=  
GaussCode[14, -16, -17, 19, 20, -22, -1, 3, 4, -6, -7, 9, 10, -12, -13, 

 15, 16, -18, -19, 21, 22, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 17, 



  18, -20, -21, 1, 2, -4, -5, 7, 8, -10, -11, 13]
In[5]:=
BR[TorusKnot[11, 3]]
Out[5]=  
BR[3, {1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 
   2}]
In[6]:=
alex = Alexander[TorusKnot[11, 3]][t]
Out[6]=  
      -10    -9    -7    -6    -4    -3   1        3    4    6    7

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



                                         t

  9    10


  t  + t
In[7]:=
Conway[TorusKnot[11, 3]][z]
Out[7]=  
        2        4         6         8         10         12

1 + 40 z  + 390 z  + 1443 z  + 2665 z  + 2782 z   + 1742 z   + 



      14        16       18    20


  666 z   + 152 z   + 19 z   + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[TorusKnot[11, 3]], KnotSignature[TorusKnot[11, 3]]}
Out[9]=  
{1, 16}
In[10]:=
J=Jones[TorusKnot[11, 3]][q]
Out[10]=  
 10    12    22
q   + q   - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[TorusKnot[11, 3]][q]
Out[12]=  
NotAvailable
In[13]:=
Kauffman[TorusKnot[11, 3]][a, z]
Out[13]=  
NotAvailable
In[14]:=
{Vassiliev[2][TorusKnot[11, 3]], Vassiliev[3][TorusKnot[11, 3]]}
Out[14]=  
{0, 220}
In[15]:=
Kh[TorusKnot[11, 3]][q, t]
Out[15]=  
 19    21    23  2    27  3    25  4    27  4    29  5    31  5

q   + q   + q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + 



  29  6    33  7    31  8    33  8    35  9    37  9    35  10
 q   t  + q   t  + q   t  + q   t  + q   t  + q   t  + q   t   + 

  39  11    37  12    39  12    41  13    43  13    41  14    45  15


  q   t   + q   t   + q   t   + q   t   + q   t   + q   t   + q   t
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