T(21,2)

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T(7,4).jpg

T(7,4)

T(11,3).jpg

T(11,3)

T(21,2).jpg See other torus knots

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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
Braid presentation
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BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gif

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]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091.

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

In[3]:= K = Knot["T(21,2)"];
In[4]:= {A = Alexander[K][t], J = Jones[K][q]}
KnotTheory::loading: Loading precomputed data in PD4Knots`.
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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], [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[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
Out[5]= {}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {}

Vassiliev invariants

V2 and V3: (55, 385)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
Data:T(21,2)/V 2,1 Data:T(21,2)/V 3,1 Data:T(21,2)/V 4,1 Data:T(21,2)/V 4,2 Data:T(21,2)/V 4,3 Data:T(21,2)/V 5,1 Data:T(21,2)/V 5,2 Data:T(21,2)/V 5,3 Data:T(21,2)/V 5,4 Data:T(21,2)/V 6,1 Data:T(21,2)/V 6,2 Data:T(21,2)/V 6,3 Data:T(21,2)/V 6,4 Data:T(21,2)/V 6,5 Data:T(21,2)/V 6,6 Data:T(21,2)/V 6,7 Data:T(21,2)/V 6,8 Data:T(21,2)/V 6,9

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

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
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=19 }[/math] [math]\displaystyle{ i=21 }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=1 }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=9 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=11 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=12 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=13 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=14 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=15 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=16 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=17 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=18 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=19 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=20 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=21 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

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

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Torus Knot Page master template (intermediate).

See/edit the Torus Knot_Splice_Base (expert).

Back to the top.

T(7,4).jpg

T(7,4)

T(11,3).jpg

T(11,3)

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