T(27,2)

T(9,4)

T(7,5)

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	T(27,2) Quick Notes

T(27,2) Further Notes and Views

Knot presentations

Planar diagram presentation	X21,49,22,48 X49,23,50,22 X23,51,24,50 X51,25,52,24 X25,53,26,52 X53,27,54,26 X27,1,28,54 X1,29,2,28 X29,3,30,2 X3,31,4,30 X31,5,32,4 X5,33,6,32 X33,7,34,6 X7,35,8,34 X35,9,36,8 X9,37,10,36 X37,11,38,10 X11,39,12,38 X39,13,40,12 X13,41,14,40 X41,15,42,14 X15,43,16,42 X43,17,44,16 X17,45,18,44 X45,19,46,18 X19,47,20,46 X47,21,48,20
Gauss code	-8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 1, -2, 3, -4, 5, -6, 7
Dowker-Thistlethwaite code	28 30 32 34 36 38 40 42 44 46 48 50 52 54 2 4 6 8 10 12 14 16 18 20 22 24 26
Conway Notation	Data:T(27,2)/Conway Notation

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^{13}-t^{12}+t^{11}-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} + t^{-11} - t^{-12} + t^{-13} }[/math]
Conway polynomial	[math]\displaystyle{ z^{26}+25 z^{24}+276 z^{22}+1771 z^{20}+7315 z^{18}+20349 z^{16}+38760 z^{14}+50388 z^{12}+43758 z^{10}+24310 z^8+8008 z^6+1365 z^4+91 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 27, 26 }
Jones polynomial	[math]\displaystyle{ -q^{40}+q^{39}-q^{38}+q^{37}-q^{36}+q^{35}-q^{34}+q^{33}-q^{32}+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^{13} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^{26}a^{-26}-26z^{24}a^{-26}-z^{24}a^{-28}+300z^{22}a^{-26}+24z^{22}a^{-28}-2024z^{20}a^{-26}-253z^{20}a^{-28}+8855z^{18}a^{-26}+1540z^{18}a^{-28}-26334z^{16}a^{-26}-5985z^{16}a^{-28}+54264z^{14}a^{-26}+15504z^{14}a^{-28}-77520z^{12}a^{-26}-27132z^{12}a^{-28}+75582z^{10}a^{-26}+31824z^{10}a^{-28}-48620z^8a^{-26}-24310z^8a^{-28}+19448z^6a^{-26}+11440z^6a^{-28}-4368z^4a^{-26}-3003z^4a^{-28}+455z^2a^{-26}+364z^2a^{-28}-14a^{-26}-13a^{-28} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{26}a^{-26}+z^{26}a^{-28}+z^{25}a^{-27}+z^{25}a^{-29}-26z^{24}a^{-26}-25z^{24}a^{-28}+z^{24}a^{-30}-24z^{23}a^{-27}-23z^{23}a^{-29}+z^{23}a^{-31}+300z^{22}a^{-26}+277z^{22}a^{-28}-22z^{22}a^{-30}+z^{22}a^{-32}+253z^{21}a^{-27}+231z^{21}a^{-29}-21z^{21}a^{-31}+z^{21}a^{-33}-2024z^{20}a^{-26}-1793z^{20}a^{-28}+210z^{20}a^{-30}-20z^{20}a^{-32}+z^{20}a^{-34}-1540z^{19}a^{-27}-1330z^{19}a^{-29}+190z^{19}a^{-31}-19z^{19}a^{-33}+z^{19}a^{-35}+8855z^{18}a^{-26}+7525z^{18}a^{-28}-1140z^{18}a^{-30}+171z^{18}a^{-32}-18z^{18}a^{-34}+z^{18}a^{-36}+5985z^{17}a^{-27}+4845z^{17}a^{-29}-969z^{17}a^{-31}+153z^{17}a^{-33}-17z^{17}a^{-35}+z^{17}a^{-37}-26334z^{16}a^{-26}-21489z^{16}a^{-28}+3876z^{16}a^{-30}-816z^{16}a^{-32}+136z^{16}a^{-34}-16z^{16}a^{-36}+z^{16}a^{-38}-15504z^{15}a^{-27}-11628z^{15}a^{-29}+3060z^{15}a^{-31}-680z^{15}a^{-33}+120z^{15}a^{-35}-15z^{15}a^{-37}+z^{15}a^{-39}+54264z^{14}a^{-26}+42636z^{14}a^{-28}-8568z^{14}a^{-30}+2380z^{14}a^{-32}-560z^{14}a^{-34}+105z^{14}a^{-36}-14z^{14}a^{-38}+z^{14}a^{-40}+27132z^{13}a^{-27}+18564z^{13}a^{-29}-6188z^{13}a^{-31}+1820z^{13}a^{-33}-455z^{13}a^{-35}+91z^{13}a^{-37}-13z^{13}a^{-39}+z^{13}a^{-41}-77520z^{12}a^{-26}-58956z^{12}a^{-28}+12376z^{12}a^{-30}-4368z^{12}a^{-32}+1365z^{12}a^{-34}-364z^{12}a^{-36}+78z^{12}a^{-38}-12z^{12}a^{-40}+z^{12}a^{-42}-31824z^{11}a^{-27}-19448z^{11}a^{-29}+8008z^{11}a^{-31}-3003z^{11}a^{-33}+1001z^{11}a^{-35}-286z^{11}a^{-37}+66z^{11}a^{-39}-11z^{11}a^{-41}+z^{11}a^{-43}+75582z^{10}a^{-26}+56134z^{10}a^{-28}-11440z^{10}a^{-30}+5005z^{10}a^{-32}-2002z^{10}a^{-34}+715z^{10}a^{-36}-220z^{10}a^{-38}+55z^{10}a^{-40}-10z^{10}a^{-42}+z^{10}a^{-44}+24310z^9a^{-27}+12870z^9a^{-29}-6435z^9a^{-31}+3003z^9a^{-33}-1287z^9a^{-35}+495z^9a^{-37}-165z^9a^{-39}+45z^9a^{-41}-9z^9a^{-43}+z^9a^{-45}-48620z^8a^{-26}-35750z^8a^{-28}+6435z^8a^{-30}-3432z^8a^{-32}+1716z^8a^{-34}-792z^8a^{-36}+330z^8a^{-38}-120z^8a^{-40}+36z^8a^{-42}-8z^8a^{-44}+z^8a^{-46}-11440z^7a^{-27}-5005z^7a^{-29}+3003z^7a^{-31}-1716z^7a^{-33}+924z^7a^{-35}-462z^7a^{-37}+210z^7a^{-39}-84z^7a^{-41}+28z^7a^{-43}-7z^7a^{-45}+z^7a^{-47}+19448z^6a^{-26}+14443z^6a^{-28}-2002z^6a^{-30}+1287z^6a^{-32}-792z^6a^{-34}+462z^6a^{-36}-252z^6a^{-38}+126z^6a^{-40}-56z^6a^{-42}+21z^6a^{-44}-6z^6a^{-46}+z^6a^{-48}+3003z^5a^{-27}+1001z^5a^{-29}-715z^5a^{-31}+495z^5a^{-33}-330z^5a^{-35}+210z^5a^{-37}-126z^5a^{-39}+70z^5a^{-41}-35z^5a^{-43}+15z^5a^{-45}-5z^5a^{-47}+z^5a^{-49}-4368z^4a^{-26}-3367z^4a^{-28}+286z^4a^{-30}-220z^4a^{-32}+165z^4a^{-34}-120z^4a^{-36}+84z^4a^{-38}-56z^4a^{-40}+35z^4a^{-42}-20z^4a^{-44}+10z^4a^{-46}-4z^4a^{-48}+z^4a^{-50}-364z^3a^{-27}-78z^3a^{-29}+66z^3a^{-31}-55z^3a^{-33}+45z^3a^{-35}-36z^3a^{-37}+28z^3a^{-39}-21z^3a^{-41}+15z^3a^{-43}-10z^3a^{-45}+6z^3a^{-47}-3z^3a^{-49}+z^3a^{-51}+455z^2a^{-26}+377z^2a^{-28}-12z^2a^{-30}+11z^2a^{-32}-10z^2a^{-34}+9z^2a^{-36}-8z^2a^{-38}+7z^2a^{-40}-6z^2a^{-42}+5z^2a^{-44}-4z^2a^{-46}+3z^2a^{-48}-2z^2a^{-50}+z^2a^{-52}+13za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-za^{-47}+za^{-49}-za^{-51}+za^{-53}-14a^{-26}-13a^{-28} }[/math]
The A2 invariant	Data:T(27,2)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(27,2)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(27,2) 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(27,2)"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ t^{13}-t^{12}+t^{11}-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} + t^{-11} - t^{-12} + t^{-13} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^{26}+25 z^{24}+276 z^{22}+1771 z^{20}+7315 z^{18}+20349 z^{16}+38760 z^{14}+50388 z^{12}+43758 z^{10}+24310 z^8+8008 z^6+1365 z^4+91 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]=

{ 27, 26 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ -q^{40}+q^{39}-q^{38}+q^{37}-q^{36}+q^{35}-q^{34}+q^{33}-q^{32}+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^{13} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ z^{26}a^{-26}-26z^{24}a^{-26}-z^{24}a^{-28}+300z^{22}a^{-26}+24z^{22}a^{-28}-2024z^{20}a^{-26}-253z^{20}a^{-28}+8855z^{18}a^{-26}+1540z^{18}a^{-28}-26334z^{16}a^{-26}-5985z^{16}a^{-28}+54264z^{14}a^{-26}+15504z^{14}a^{-28}-77520z^{12}a^{-26}-27132z^{12}a^{-28}+75582z^{10}a^{-26}+31824z^{10}a^{-28}-48620z^8a^{-26}-24310z^8a^{-28}+19448z^6a^{-26}+11440z^6a^{-28}-4368z^4a^{-26}-3003z^4a^{-28}+455z^2a^{-26}+364z^2a^{-28}-14a^{-26}-13a^{-28} }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ z^{26}a^{-26}+z^{26}a^{-28}+z^{25}a^{-27}+z^{25}a^{-29}-26z^{24}a^{-26}-25z^{24}a^{-28}+z^{24}a^{-30}-24z^{23}a^{-27}-23z^{23}a^{-29}+z^{23}a^{-31}+300z^{22}a^{-26}+277z^{22}a^{-28}-22z^{22}a^{-30}+z^{22}a^{-32}+253z^{21}a^{-27}+231z^{21}a^{-29}-21z^{21}a^{-31}+z^{21}a^{-33}-2024z^{20}a^{-26}-1793z^{20}a^{-28}+210z^{20}a^{-30}-20z^{20}a^{-32}+z^{20}a^{-34}-1540z^{19}a^{-27}-1330z^{19}a^{-29}+190z^{19}a^{-31}-19z^{19}a^{-33}+z^{19}a^{-35}+8855z^{18}a^{-26}+7525z^{18}a^{-28}-1140z^{18}a^{-30}+171z^{18}a^{-32}-18z^{18}a^{-34}+z^{18}a^{-36}+5985z^{17}a^{-27}+4845z^{17}a^{-29}-969z^{17}a^{-31}+153z^{17}a^{-33}-17z^{17}a^{-35}+z^{17}a^{-37}-26334z^{16}a^{-26}-21489z^{16}a^{-28}+3876z^{16}a^{-30}-816z^{16}a^{-32}+136z^{16}a^{-34}-16z^{16}a^{-36}+z^{16}a^{-38}-15504z^{15}a^{-27}-11628z^{15}a^{-29}+3060z^{15}a^{-31}-680z^{15}a^{-33}+120z^{15}a^{-35}-15z^{15}a^{-37}+z^{15}a^{-39}+54264z^{14}a^{-26}+42636z^{14}a^{-28}-8568z^{14}a^{-30}+2380z^{14}a^{-32}-560z^{14}a^{-34}+105z^{14}a^{-36}-14z^{14}a^{-38}+z^{14}a^{-40}+27132z^{13}a^{-27}+18564z^{13}a^{-29}-6188z^{13}a^{-31}+1820z^{13}a^{-33}-455z^{13}a^{-35}+91z^{13}a^{-37}-13z^{13}a^{-39}+z^{13}a^{-41}-77520z^{12}a^{-26}-58956z^{12}a^{-28}+12376z^{12}a^{-30}-4368z^{12}a^{-32}+1365z^{12}a^{-34}-364z^{12}a^{-36}+78z^{12}a^{-38}-12z^{12}a^{-40}+z^{12}a^{-42}-31824z^{11}a^{-27}-19448z^{11}a^{-29}+8008z^{11}a^{-31}-3003z^{11}a^{-33}+1001z^{11}a^{-35}-286z^{11}a^{-37}+66z^{11}a^{-39}-11z^{11}a^{-41}+z^{11}a^{-43}+75582z^{10}a^{-26}+56134z^{10}a^{-28}-11440z^{10}a^{-30}+5005z^{10}a^{-32}-2002z^{10}a^{-34}+715z^{10}a^{-36}-220z^{10}a^{-38}+55z^{10}a^{-40}-10z^{10}a^{-42}+z^{10}a^{-44}+24310z^9a^{-27}+12870z^9a^{-29}-6435z^9a^{-31}+3003z^9a^{-33}-1287z^9a^{-35}+495z^9a^{-37}-165z^9a^{-39}+45z^9a^{-41}-9z^9a^{-43}+z^9a^{-45}-48620z^8a^{-26}-35750z^8a^{-28}+6435z^8a^{-30}-3432z^8a^{-32}+1716z^8a^{-34}-792z^8a^{-36}+330z^8a^{-38}-120z^8a^{-40}+36z^8a^{-42}-8z^8a^{-44}+z^8a^{-46}-11440z^7a^{-27}-5005z^7a^{-29}+3003z^7a^{-31}-1716z^7a^{-33}+924z^7a^{-35}-462z^7a^{-37}+210z^7a^{-39}-84z^7a^{-41}+28z^7a^{-43}-7z^7a^{-45}+z^7a^{-47}+19448z^6a^{-26}+14443z^6a^{-28}-2002z^6a^{-30}+1287z^6a^{-32}-792z^6a^{-34}+462z^6a^{-36}-252z^6a^{-38}+126z^6a^{-40}-56z^6a^{-42}+21z^6a^{-44}-6z^6a^{-46}+z^6a^{-48}+3003z^5a^{-27}+1001z^5a^{-29}-715z^5a^{-31}+495z^5a^{-33}-330z^5a^{-35}+210z^5a^{-37}-126z^5a^{-39}+70z^5a^{-41}-35z^5a^{-43}+15z^5a^{-45}-5z^5a^{-47}+z^5a^{-49}-4368z^4a^{-26}-3367z^4a^{-28}+286z^4a^{-30}-220z^4a^{-32}+165z^4a^{-34}-120z^4a^{-36}+84z^4a^{-38}-56z^4a^{-40}+35z^4a^{-42}-20z^4a^{-44}+10z^4a^{-46}-4z^4a^{-48}+z^4a^{-50}-364z^3a^{-27}-78z^3a^{-29}+66z^3a^{-31}-55z^3a^{-33}+45z^3a^{-35}-36z^3a^{-37}+28z^3a^{-39}-21z^3a^{-41}+15z^3a^{-43}-10z^3a^{-45}+6z^3a^{-47}-3z^3a^{-49}+z^3a^{-51}+455z^2a^{-26}+377z^2a^{-28}-12z^2a^{-30}+11z^2a^{-32}-10z^2a^{-34}+9z^2a^{-36}-8z^2a^{-38}+7z^2a^{-40}-6z^2a^{-42}+5z^2a^{-44}-4z^2a^{-46}+3z^2a^{-48}-2z^2a^{-50}+z^2a^{-52}+13za^{-27}+za^{-29}-za^{-31}+za^{-33}-za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-za^{-47}+za^{-49}-za^{-51}+za^{-53}-14a^{-26}-13a^{-28} }[/math]

Vassiliev invariants

V2 and V3:

(91, 819)

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(27,2)/V 2,1 Data:T(27,2)/V 3,1 Data:T(27,2)/V 4,1 Data:T(27,2)/V 4,2 Data:T(27,2)/V 4,3 Data:T(27,2)/V 5,1 Data:T(27,2)/V 5,2 Data:T(27,2)/V 5,3 Data:T(27,2)/V 5,4 Data:T(27,2)/V 6,1 Data:T(27,2)/V 6,2 Data:T(27,2)/V 6,3 Data:T(27,2)/V 6,4 Data:T(27,2)/V 6,5 Data:T(27,2)/V 6,6 Data:T(27,2)/V 6,7 Data:T(27,2)/V 6,8 Data:T(27,2)/V 6,9

V_2,1 through V_6,9 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]26 is the signature of T(27,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 χ 81                                                       1 -1 79                                                         0 77                                                   1 1   0 75                                                         0 73                                               1 1       0 71                                                         0 69                                           1 1           0 67                                                         0 65                                       1 1               0 63                                                         0 61                                   1 1                   0 59                                                         0 57                               1 1                       0 55                                                         0 53                           1 1                           0 51                                                         0 49                       1 1                               0 47                                                         0 45                   1 1                                   0 43                                                         0 41               1 1                                       0 39                                                         0 37           1 1                                           0 35                                                         0 33       1 1                                               0 31                                                         0 29     1                                                   1 27 1                                                       1 25 1                                                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=25 }[/math] [math]\displaystyle{ i=27 }[/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] [math]\displaystyle{ r=22 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=23 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=24 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=25 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=26 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=27 }[/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.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[TorusKnot[27, 2]]
Out[2]=	27
In[3]:=	PD[TorusKnot[27, 2]]
Out[3]=	PD[X[21, 49, 22, 48], X[49, 23, 50, 22], X[23, 51, 24, 50], X[51, 25, 52, 24], X[25, 53, 26, 52], X[53, 27, 54, 26], X[27, 1, 28, 54], X[1, 29, 2, 28], X[29, 3, 30, 2], X[3, 31, 4, 30], X[31, 5, 32, 4], X[5, 33, 6, 32], X[33, 7, 34, 6], X[7, 35, 8, 34], X[35, 9, 36, 8], X[9, 37, 10, 36], X[37, 11, 38, 10], X[11, 39, 12, 38], X[39, 13, 40, 12], X[13, 41, 14, 40], X[41, 15, 42, 14], X[15, 43, 16, 42], X[43, 17, 44, 16], X[17, 45, 18, 44], X[45, 19, 46, 18], X[19, 47, 20, 46], X[47, 21, 48, 20]]
In[4]:=	GaussCode[TorusKnot[27, 2]]
Out[4]=	GaussCode[-8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 1, -2, 3, -4, 5, -6, 7]
In[5]:=	BR[TorusKnot[27, 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, 1, 1, 1, 1, 1, 1}]
In[6]:=	alex = Alexander[TorusKnot[27, 2]][t]
Out[6]=	-13 -12 -11 -1 + Alternating - Alternating + Alternating - -10 -9 -8 -7 Alternating + Alternating - Alternating + Alternating - -6 -5 -4 -3 Alternating + Alternating - Alternating + Alternating - -2 1 2 Alternating + ----------- + Alternating - Alternating + Alternating 3 4 5 6 Alternating - Alternating + Alternating - Alternating + 7 8 9 10 Alternating - Alternating + Alternating - Alternating + 11 12 13 Alternating - Alternating + Alternating
In[7]:=	Conway[TorusKnot[27, 2]][z]
Out[7]=	2 4 6 8 10 12 1 + 91 z + 1365 z + 8008 z + 24310 z + 43758 z + 50388 z + 14 16 18 20 22 24 26 38760 z + 20349 z + 7315 z + 1771 z + 276 z + 25 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[TorusKnot[27, 2]], KnotSignature[TorusKnot[27, 2]]}
Out[9]=	{27, 26}
In[10]:=	J=Jones[TorusKnot[27, 2]][q]
Out[10]=	13 15 16 17 18 19 20 21 22 23 24 25 q + q - q + q - q + q - q + q - q + q - q + q - 26 27 28 29 30 31 32 33 34 35 36 q + q - q + q - q + q - q + q - q + q - q + 37 38 39 40 q - q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{}
In[12]:=	A2Invariant[TorusKnot[27, 2]][q]
Out[12]=	NotAvailable
In[13]:=	Kauffman[TorusKnot[27, 2]][a, z]
Out[13]=	NotAvailable
In[14]:=	{Vassiliev[2][TorusKnot[27, 2]], Vassiliev[3][TorusKnot[27, 2]]}
Out[14]=	{0, 819}
In[15]:=	Kh[TorusKnot[27, 2]][q, t]
Out[15]=	25 27 2 29 3 33 4 33 q + q + Alternating q + Alternating q + Alternating q + 5 37 6 37 7 41 Alternating q + Alternating q + Alternating q + 8 41 9 45 10 45 Alternating q + Alternating q + Alternating q + 11 49 12 49 13 53 Alternating q + Alternating q + Alternating q + 14 53 15 57 16 57 Alternating q + Alternating q + Alternating q + 17 61 18 61 19 65 Alternating q + Alternating q + Alternating q + 20 65 21 69 22 69 Alternating q + Alternating q + Alternating q + 23 73 24 73 25 77 Alternating q + Alternating q + Alternating q + 26 77 27 81 Alternating q + Alternating q