T(21,2)

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[[Image:T(7,4).{{{ext}}}|80px|link=T(7,4)]]

T(7,4)

[[Image:T(11,3).{{{ext}}}|80px|link=T(11,3)]]

T(11,3)

Visit T(21,2)'s page at Knotilus!

Visit T(21,2)'s page at the original Knot Atlas!

Knot presentations

Planar diagram presentation X11,33,12,32 X33,13,34,12 X13,35,14,34 X35,15,36,14 X15,37,16,36 X37,17,38,16 X17,39,18,38 X39,19,40,18 X19,41,20,40 X41,21,42,20 X21,1,22,42 X1,23,2,22 X23,3,24,2 X3,25,4,24 X25,5,26,4 X5,27,6,26 X27,7,28,6 X7,29,8,28 X29,9,30,8 X9,31,10,30 X31,11,32,10
Gauss code {-12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11}
Dowker-Thistlethwaite code 22 24 26 28 30 32 34 36 38 40 42 2 4 6 8 10 12 14 16 18 20

Polynomial invariants

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^{10}-t^9+t^8-t^7+t^6-t^5+t^4-t^3+t^2-t+1- t^{-1} + t^{-2} - t^{-3} + t^{-4} - t^{-5} + t^{-6} - t^{-7} + t^{-8} - t^{-9} + t^{-10} }[/math]
Conway polynomial [math]\displaystyle{ z^{20}+19 z^{18}+153 z^{16}+680 z^{14}+1820 z^{12}+3003 z^{10}+3003 z^8+1716 z^6+495 z^4+55 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 21, 20 }
Jones polynomial [math]\displaystyle{ -q^{31}+q^{30}-q^{29}+q^{28}-q^{27}+q^{26}-q^{25}+q^{24}-q^{23}+q^{22}-q^{21}+q^{20}-q^{19}+q^{18}-q^{17}+q^{16}-q^{15}+q^{14}-q^{13}+q^{12}+q^{10} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^{20} a^{-20} +20 z^{18} a^{-20} -z^{18} a^{-22} +171 z^{16} a^{-20} -18 z^{16} a^{-22} +816 z^{14} a^{-20} -136 z^{14} a^{-22} +2380 z^{12} a^{-20} -560 z^{12} a^{-22} +4368 z^{10} a^{-20} -1365 z^{10} a^{-22} +5005 z^8 a^{-20} -2002 z^8 a^{-22} +3432 z^6 a^{-20} -1716 z^6 a^{-22} +1287 z^4 a^{-20} -792 z^4 a^{-22} +220 z^2 a^{-20} -165 z^2 a^{-22} +11 a^{-20} -10 a^{-22} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^{20}a^{-20}+z^{20}a^{-22}+z^{19}a^{-21}+z^{19}a^{-23}-20z^{18}a^{-20}-19z^{18}a^{-22}+z^{18}a^{-24}-18z^{17}a^{-21}-17z^{17}a^{-23}+z^{17}a^{-25}+171z^{16}a^{-20}+154z^{16}a^{-22}-16z^{16}a^{-24}+z^{16}a^{-26}+136z^{15}a^{-21}+120z^{15}a^{-23}-15z^{15}a^{-25}+z^{15}a^{-27}-816z^{14}a^{-20}-696z^{14}a^{-22}+105z^{14}a^{-24}-14z^{14}a^{-26}+z^{14}a^{-28}-560z^{13}a^{-21}-455z^{13}a^{-23}+91z^{13}a^{-25}-13z^{13}a^{-27}+z^{13}a^{-29}+2380z^{12}a^{-20}+1925z^{12}a^{-22}-364z^{12}a^{-24}+78z^{12}a^{-26}-12z^{12}a^{-28}+z^{12}a^{-30}+1365z^{11}a^{-21}+1001z^{11}a^{-23}-286z^{11}a^{-25}+66z^{11}a^{-27}-11z^{11}a^{-29}+z^{11}a^{-31}-4368z^{10}a^{-20}-3367z^{10}a^{-22}+715z^{10}a^{-24}-220z^{10}a^{-26}+55z^{10}a^{-28}-10z^{10}a^{-30}+z^{10}a^{-32}-2002z^9a^{-21}-1287z^9a^{-23}+495z^9a^{-25}-165z^9a^{-27}+45z^9a^{-29}-9z^9a^{-31}+z^9a^{-33}+5005z^8a^{-20}+3718z^8a^{-22}-792z^8a^{-24}+330z^8a^{-26}-120z^8a^{-28}+36z^8a^{-30}-8z^8a^{-32}+z^8a^{-34}+1716z^7a^{-21}+924z^7a^{-23}-462z^7a^{-25}+210z^7a^{-27}-84z^7a^{-29}+28z^7a^{-31}-7z^7a^{-33}+z^7a^{-35}-3432z^6a^{-20}-2508z^6a^{-22}+462z^6a^{-24}-252z^6a^{-26}+126z^6a^{-28}-56z^6a^{-30}+21z^6a^{-32}-6z^6a^{-34}+z^6a^{-36}-792z^5a^{-21}-330z^5a^{-23}+210z^5a^{-25}-126z^5a^{-27}+70z^5a^{-29}-35z^5a^{-31}+15z^5a^{-33}-5z^5a^{-35}+z^5a^{-37}+1287z^4a^{-20}+957z^4a^{-22}-120z^4a^{-24}+84z^4a^{-26}-56z^4a^{-28}+35z^4a^{-30}-20z^4a^{-32}+10z^4a^{-34}-4z^4a^{-36}+z^4a^{-38}+165z^3a^{-21}+45z^3a^{-23}-36z^3a^{-25}+28z^3a^{-27}-21z^3a^{-29}+15z^3a^{-31}-10z^3a^{-33}+6z^3a^{-35}-3z^3a^{-37}+z^3a^{-39}-220z^2a^{-20}-175z^2a^{-22}+9z^2a^{-24}-8z^2a^{-26}+7z^2a^{-28}-6z^2a^{-30}+5z^2a^{-32}-4z^2a^{-34}+3z^2a^{-36}-2z^2a^{-38}+z^2a^{-40}-10za^{-21}-za^{-23}+za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}+11a^{-20}+10a^{-22} }[/math]
The A2 invariant Data:T(21,2)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(21,2)/QuantumInvariant/G2/1,0
Further Quantum Invariants
T(21,2) 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(21,2)"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^{10}-t^9+t^8-t^7+t^6-t^5+t^4-t^3+t^2-t+1- t^{-1} + t^{-2} - t^{-3} + t^{-4} - t^{-5} + t^{-6} - t^{-7} + t^{-8} - t^{-9} + t^{-10} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^{20}+19 z^{18}+153 z^{16}+680 z^{14}+1820 z^{12}+3003 z^{10}+3003 z^8+1716 z^6+495 z^4+55 z^2+1 }[/math]
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]= [math]\displaystyle{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 21, 20 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^{31}+q^{30}-q^{29}+q^{28}-q^{27}+q^{26}-q^{25}+q^{24}-q^{23}+q^{22}-q^{21}+q^{20}-q^{19}+q^{18}-q^{17}+q^{16}-q^{15}+q^{14}-q^{13}+q^{12}+q^{10} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^{20} a^{-20} +20 z^{18} a^{-20} -z^{18} a^{-22} +171 z^{16} a^{-20} -18 z^{16} a^{-22} +816 z^{14} a^{-20} -136 z^{14} a^{-22} +2380 z^{12} a^{-20} -560 z^{12} a^{-22} +4368 z^{10} a^{-20} -1365 z^{10} a^{-22} +5005 z^8 a^{-20} -2002 z^8 a^{-22} +3432 z^6 a^{-20} -1716 z^6 a^{-22} +1287 z^4 a^{-20} -792 z^4 a^{-22} +220 z^2 a^{-20} -165 z^2 a^{-22} +11 a^{-20} -10 a^{-22} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^{20}a^{-20}+z^{20}a^{-22}+z^{19}a^{-21}+z^{19}a^{-23}-20z^{18}a^{-20}-19z^{18}a^{-22}+z^{18}a^{-24}-18z^{17}a^{-21}-17z^{17}a^{-23}+z^{17}a^{-25}+171z^{16}a^{-20}+154z^{16}a^{-22}-16z^{16}a^{-24}+z^{16}a^{-26}+136z^{15}a^{-21}+120z^{15}a^{-23}-15z^{15}a^{-25}+z^{15}a^{-27}-816z^{14}a^{-20}-696z^{14}a^{-22}+105z^{14}a^{-24}-14z^{14}a^{-26}+z^{14}a^{-28}-560z^{13}a^{-21}-455z^{13}a^{-23}+91z^{13}a^{-25}-13z^{13}a^{-27}+z^{13}a^{-29}+2380z^{12}a^{-20}+1925z^{12}a^{-22}-364z^{12}a^{-24}+78z^{12}a^{-26}-12z^{12}a^{-28}+z^{12}a^{-30}+1365z^{11}a^{-21}+1001z^{11}a^{-23}-286z^{11}a^{-25}+66z^{11}a^{-27}-11z^{11}a^{-29}+z^{11}a^{-31}-4368z^{10}a^{-20}-3367z^{10}a^{-22}+715z^{10}a^{-24}-220z^{10}a^{-26}+55z^{10}a^{-28}-10z^{10}a^{-30}+z^{10}a^{-32}-2002z^9a^{-21}-1287z^9a^{-23}+495z^9a^{-25}-165z^9a^{-27}+45z^9a^{-29}-9z^9a^{-31}+z^9a^{-33}+5005z^8a^{-20}+3718z^8a^{-22}-792z^8a^{-24}+330z^8a^{-26}-120z^8a^{-28}+36z^8a^{-30}-8z^8a^{-32}+z^8a^{-34}+1716z^7a^{-21}+924z^7a^{-23}-462z^7a^{-25}+210z^7a^{-27}-84z^7a^{-29}+28z^7a^{-31}-7z^7a^{-33}+z^7a^{-35}-3432z^6a^{-20}-2508z^6a^{-22}+462z^6a^{-24}-252z^6a^{-26}+126z^6a^{-28}-56z^6a^{-30}+21z^6a^{-32}-6z^6a^{-34}+z^6a^{-36}-792z^5a^{-21}-330z^5a^{-23}+210z^5a^{-25}-126z^5a^{-27}+70z^5a^{-29}-35z^5a^{-31}+15z^5a^{-33}-5z^5a^{-35}+z^5a^{-37}+1287z^4a^{-20}+957z^4a^{-22}-120z^4a^{-24}+84z^4a^{-26}-56z^4a^{-28}+35z^4a^{-30}-20z^4a^{-32}+10z^4a^{-34}-4z^4a^{-36}+z^4a^{-38}+165z^3a^{-21}+45z^3a^{-23}-36z^3a^{-25}+28z^3a^{-27}-21z^3a^{-29}+15z^3a^{-31}-10z^3a^{-33}+6z^3a^{-35}-3z^3a^{-37}+z^3a^{-39}-220z^2a^{-20}-175z^2a^{-22}+9z^2a^{-24}-8z^2a^{-26}+7z^2a^{-28}-6z^2a^{-30}+5z^2a^{-32}-4z^2a^{-34}+3z^2a^{-36}-2z^2a^{-38}+z^2a^{-40}-10za^{-21}-za^{-23}+za^{-25}-za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}+11a^{-20}+10a^{-22} }[/math]

Vassiliev invariants

V2 and V3 {0, 385})

Khovanov Homology. The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s+1 }[/math], where [math]\displaystyle{ s= }[/math]20 is the signature of T(21,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 0123456789101112131415161718192021χ
63                     1-1
61                      0
59                   11 0
57                      0
55                 11   0
53                      0
51               11     0
49                      0
47             11       0
45                      0
43           11         0
41                      0
39         11           0
37                      0
35       11             0
33                      0
31     11               0
29                      0
27   11                 0
25                      0
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[21, 2]]
Out[2]=   
21
In[3]:=
PD[TorusKnot[21, 2]]
Out[3]=   
PD[X[11, 33, 12, 32], X[33, 13, 34, 12], X[13, 35, 14, 34], 

 X[35, 15, 36, 14], X[15, 37, 16, 36], X[37, 17, 38, 16], 

 X[17, 39, 18, 38], X[39, 19, 40, 18], X[19, 41, 20, 40], 

 X[41, 21, 42, 20], X[21, 1, 22, 42], X[1, 23, 2, 22], 

 X[23, 3, 24, 2], X[3, 25, 4, 24], X[25, 5, 26, 4], X[5, 27, 6, 26], 

 X[27, 7, 28, 6], X[7, 29, 8, 28], X[29, 9, 30, 8], X[9, 31, 10, 30], 



  X[31, 11, 32, 10]]
In[4]:=
GaussCode[TorusKnot[21, 2]]
Out[4]=   
GaussCode[-12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -1, 2, -3, 4, 

 -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, 



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

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



                                                          t

  3    4    5    6    7    8    9    10


  t  + t  - t  + t  - t  + t  - t  + t
In[7]:=
Conway[TorusKnot[21, 2]][z]
Out[7]=   
        2        4         6         8         10         12

1 + 55 z  + 495 z  + 1716 z  + 3003 z  + 3003 z   + 1820 z   + 



      14        16       18    20


  680 z   + 153 z   + 19 z   + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{}
In[9]:=
{KnotDet[TorusKnot[21, 2]], KnotSignature[TorusKnot[21, 2]]}
Out[9]=   
{21, 20}
In[10]:=
J=Jones[TorusKnot[21, 2]][q]
Out[10]=   
 10    12    13    14    15    16    17    18    19    20    21    22

q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - q   + q   - 



  23    24    25    26    27    28    29    30    31


  q   + q   - q   + q   - q   + q   - q   + q   - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{}
In[12]:=
A2Invariant[TorusKnot[21, 2]][q]
Out[12]=   
NotAvailable
In[13]:=
Kauffman[TorusKnot[21, 2]][a, z]
Out[13]=   
NotAvailable
In[14]:=
{Vassiliev[2][TorusKnot[21, 2]], Vassiliev[3][TorusKnot[21, 2]]}
Out[14]=   
{0, 385}
In[15]:=
Kh[TorusKnot[21, 2]][q, t]
Out[15]=   
 19    21    23  2    27  3    27  4    31  5    31  6    35  7

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



  35  8    39  9    39  10    43  11    43  12    47  13    47  14
 q   t  + q   t  + q   t   + q   t   + q   t   + q   t   + q   t   + 

  51  15    51  16    55  17    55  18    59  19    59  20    63  21


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