T(29,2)

T(14,3)

T(31,2)

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	T(29,2) Quick Notes

T(29,2) Further Notes and Views

Knot presentations

Planar diagram presentation	X19,49,20,48 X49,21,50,20 X21,51,22,50 X51,23,52,22 X23,53,24,52 X53,25,54,24 X25,55,26,54 X55,27,56,26 X27,57,28,56 X57,29,58,28 X29,1,30,58 X1,31,2,30 X31,3,32,2 X3,33,4,32 X33,5,34,4 X5,35,6,34 X35,7,36,6 X7,37,8,36 X37,9,38,8 X9,39,10,38 X39,11,40,10 X11,41,12,40 X41,13,42,12 X13,43,14,42 X43,15,44,14 X15,45,16,44 X45,17,46,16 X17,47,18,46 X47,19,48,18
Gauss code	-12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -28, 29, -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, 28, -29, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11
Dowker-Thistlethwaite code	30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 2 4 6 8 10 12 14 16 18 20 22 24 26 28
Conway Notation	Data:T(29,2)/Conway Notation

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^{14}-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} + t^{-14} }[/math]
Conway polynomial	[math]\displaystyle{ z^{28}+27 z^{26}+325 z^{24}+2300 z^{22}+10626 z^{20}+33649 z^{18}+74613 z^{16}+116280 z^{14}+125970 z^{12}+92378 z^{10}+43758 z^8+12376 z^6+1820 z^4+105 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 29, 28 }
Jones polynomial	[math]\displaystyle{ -q^{43}+q^{42}-q^{41}+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^{14} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^{28}a^{-28}-28z^{26}a^{-28}-z^{26}a^{-30}+351z^{24}a^{-28}+26z^{24}a^{-30}-2600z^{22}a^{-28}-300z^{22}a^{-30}+12650z^{20}a^{-28}+2024z^{20}a^{-30}-42504z^{18}a^{-28}-8855z^{18}a^{-30}+100947z^{16}a^{-28}+26334z^{16}a^{-30}-170544z^{14}a^{-28}-54264z^{14}a^{-30}+203490z^{12}a^{-28}+77520z^{12}a^{-30}-167960z^{10}a^{-28}-75582z^{10}a^{-30}+92378z^8a^{-28}+48620z^8a^{-30}-31824z^6a^{-28}-19448z^6a^{-30}+6188z^4a^{-28}+4368z^4a^{-30}-560z^2a^{-28}-455z^2a^{-30}+15a^{-28}+14a^{-30} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{28}a^{-28}+z^{28}a^{-30}+z^{27}a^{-29}+z^{27}a^{-31}-28z^{26}a^{-28}-27z^{26}a^{-30}+z^{26}a^{-32}-26z^{25}a^{-29}-25z^{25}a^{-31}+z^{25}a^{-33}+351z^{24}a^{-28}+326z^{24}a^{-30}-24z^{24}a^{-32}+z^{24}a^{-34}+300z^{23}a^{-29}+276z^{23}a^{-31}-23z^{23}a^{-33}+z^{23}a^{-35}-2600z^{22}a^{-28}-2324z^{22}a^{-30}+253z^{22}a^{-32}-22z^{22}a^{-34}+z^{22}a^{-36}-2024z^{21}a^{-29}-1771z^{21}a^{-31}+231z^{21}a^{-33}-21z^{21}a^{-35}+z^{21}a^{-37}+12650z^{20}a^{-28}+10879z^{20}a^{-30}-1540z^{20}a^{-32}+210z^{20}a^{-34}-20z^{20}a^{-36}+z^{20}a^{-38}+8855z^{19}a^{-29}+7315z^{19}a^{-31}-1330z^{19}a^{-33}+190z^{19}a^{-35}-19z^{19}a^{-37}+z^{19}a^{-39}-42504z^{18}a^{-28}-35189z^{18}a^{-30}+5985z^{18}a^{-32}-1140z^{18}a^{-34}+171z^{18}a^{-36}-18z^{18}a^{-38}+z^{18}a^{-40}-26334z^{17}a^{-29}-20349z^{17}a^{-31}+4845z^{17}a^{-33}-969z^{17}a^{-35}+153z^{17}a^{-37}-17z^{17}a^{-39}+z^{17}a^{-41}+100947z^{16}a^{-28}+80598z^{16}a^{-30}-15504z^{16}a^{-32}+3876z^{16}a^{-34}-816z^{16}a^{-36}+136z^{16}a^{-38}-16z^{16}a^{-40}+z^{16}a^{-42}+54264z^{15}a^{-29}+38760z^{15}a^{-31}-11628z^{15}a^{-33}+3060z^{15}a^{-35}-680z^{15}a^{-37}+120z^{15}a^{-39}-15z^{15}a^{-41}+z^{15}a^{-43}-170544z^{14}a^{-28}-131784z^{14}a^{-30}+27132z^{14}a^{-32}-8568z^{14}a^{-34}+2380z^{14}a^{-36}-560z^{14}a^{-38}+105z^{14}a^{-40}-14z^{14}a^{-42}+z^{14}a^{-44}-77520z^{13}a^{-29}-50388z^{13}a^{-31}+18564z^{13}a^{-33}-6188z^{13}a^{-35}+1820z^{13}a^{-37}-455z^{13}a^{-39}+91z^{13}a^{-41}-13z^{13}a^{-43}+z^{13}a^{-45}+203490z^{12}a^{-28}+153102z^{12}a^{-30}-31824z^{12}a^{-32}+12376z^{12}a^{-34}-4368z^{12}a^{-36}+1365z^{12}a^{-38}-364z^{12}a^{-40}+78z^{12}a^{-42}-12z^{12}a^{-44}+z^{12}a^{-46}+75582z^{11}a^{-29}+43758z^{11}a^{-31}-19448z^{11}a^{-33}+8008z^{11}a^{-35}-3003z^{11}a^{-37}+1001z^{11}a^{-39}-286z^{11}a^{-41}+66z^{11}a^{-43}-11z^{11}a^{-45}+z^{11}a^{-47}-167960z^{10}a^{-28}-124202z^{10}a^{-30}+24310z^{10}a^{-32}-11440z^{10}a^{-34}+5005z^{10}a^{-36}-2002z^{10}a^{-38}+715z^{10}a^{-40}-220z^{10}a^{-42}+55z^{10}a^{-44}-10z^{10}a^{-46}+z^{10}a^{-48}-48620z^9a^{-29}-24310z^9a^{-31}+12870z^9a^{-33}-6435z^9a^{-35}+3003z^9a^{-37}-1287z^9a^{-39}+495z^9a^{-41}-165z^9a^{-43}+45z^9a^{-45}-9z^9a^{-47}+z^9a^{-49}+92378z^8a^{-28}+68068z^8a^{-30}-11440z^8a^{-32}+6435z^8a^{-34}-3432z^8a^{-36}+1716z^8a^{-38}-792z^8a^{-40}+330z^8a^{-42}-120z^8a^{-44}+36z^8a^{-46}-8z^8a^{-48}+z^8a^{-50}+19448z^7a^{-29}+8008z^7a^{-31}-5005z^7a^{-33}+3003z^7a^{-35}-1716z^7a^{-37}+924z^7a^{-39}-462z^7a^{-41}+210z^7a^{-43}-84z^7a^{-45}+28z^7a^{-47}-7z^7a^{-49}+z^7a^{-51}-31824z^6a^{-28}-23816z^6a^{-30}+3003z^6a^{-32}-2002z^6a^{-34}+1287z^6a^{-36}-792z^6a^{-38}+462z^6a^{-40}-252z^6a^{-42}+126z^6a^{-44}-56z^6a^{-46}+21z^6a^{-48}-6z^6a^{-50}+z^6a^{-52}-4368z^5a^{-29}-1365z^5a^{-31}+1001z^5a^{-33}-715z^5a^{-35}+495z^5a^{-37}-330z^5a^{-39}+210z^5a^{-41}-126z^5a^{-43}+70z^5a^{-45}-35z^5a^{-47}+15z^5a^{-49}-5z^5a^{-51}+z^5a^{-53}+6188z^4a^{-28}+4823z^4a^{-30}-364z^4a^{-32}+286z^4a^{-34}-220z^4a^{-36}+165z^4a^{-38}-120z^4a^{-40}+84z^4a^{-42}-56z^4a^{-44}+35z^4a^{-46}-20z^4a^{-48}+10z^4a^{-50}-4z^4a^{-52}+z^4a^{-54}+455z^3a^{-29}+91z^3a^{-31}-78z^3a^{-33}+66z^3a^{-35}-55z^3a^{-37}+45z^3a^{-39}-36z^3a^{-41}+28z^3a^{-43}-21z^3a^{-45}+15z^3a^{-47}-10z^3a^{-49}+6z^3a^{-51}-3z^3a^{-53}+z^3a^{-55}-560z^2a^{-28}-469z^2a^{-30}+13z^2a^{-32}-12z^2a^{-34}+11z^2a^{-36}-10z^2a^{-38}+9z^2a^{-40}-8z^2a^{-42}+7z^2a^{-44}-6z^2a^{-46}+5z^2a^{-48}-4z^2a^{-50}+3z^2a^{-52}-2z^2a^{-54}+z^2a^{-56}-14za^{-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}-za^{-55}+za^{-57}+15a^{-28}+14a^{-30} }[/math]
The A2 invariant	Data:T(29,2)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(29,2)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(29,2) 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(29,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^{14}-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} + t^{-14} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^{28}+27 z^{26}+325 z^{24}+2300 z^{22}+10626 z^{20}+33649 z^{18}+74613 z^{16}+116280 z^{14}+125970 z^{12}+92378 z^{10}+43758 z^8+12376 z^6+1820 z^4+105 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]=

{ 29, 28 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^{43}+q^{42}-q^{41}+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^{14} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^{28}a^{-28}-28z^{26}a^{-28}-z^{26}a^{-30}+351z^{24}a^{-28}+26z^{24}a^{-30}-2600z^{22}a^{-28}-300z^{22}a^{-30}+12650z^{20}a^{-28}+2024z^{20}a^{-30}-42504z^{18}a^{-28}-8855z^{18}a^{-30}+100947z^{16}a^{-28}+26334z^{16}a^{-30}-170544z^{14}a^{-28}-54264z^{14}a^{-30}+203490z^{12}a^{-28}+77520z^{12}a^{-30}-167960z^{10}a^{-28}-75582z^{10}a^{-30}+92378z^8a^{-28}+48620z^8a^{-30}-31824z^6a^{-28}-19448z^6a^{-30}+6188z^4a^{-28}+4368z^4a^{-30}-560z^2a^{-28}-455z^2a^{-30}+15a^{-28}+14a^{-30} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^{28}a^{-28}+z^{28}a^{-30}+z^{27}a^{-29}+z^{27}a^{-31}-28z^{26}a^{-28}-27z^{26}a^{-30}+z^{26}a^{-32}-26z^{25}a^{-29}-25z^{25}a^{-31}+z^{25}a^{-33}+351z^{24}a^{-28}+326z^{24}a^{-30}-24z^{24}a^{-32}+z^{24}a^{-34}+300z^{23}a^{-29}+276z^{23}a^{-31}-23z^{23}a^{-33}+z^{23}a^{-35}-2600z^{22}a^{-28}-2324z^{22}a^{-30}+253z^{22}a^{-32}-22z^{22}a^{-34}+z^{22}a^{-36}-2024z^{21}a^{-29}-1771z^{21}a^{-31}+231z^{21}a^{-33}-21z^{21}a^{-35}+z^{21}a^{-37}+12650z^{20}a^{-28}+10879z^{20}a^{-30}-1540z^{20}a^{-32}+210z^{20}a^{-34}-20z^{20}a^{-36}+z^{20}a^{-38}+8855z^{19}a^{-29}+7315z^{19}a^{-31}-1330z^{19}a^{-33}+190z^{19}a^{-35}-19z^{19}a^{-37}+z^{19}a^{-39}-42504z^{18}a^{-28}-35189z^{18}a^{-30}+5985z^{18}a^{-32}-1140z^{18}a^{-34}+171z^{18}a^{-36}-18z^{18}a^{-38}+z^{18}a^{-40}-26334z^{17}a^{-29}-20349z^{17}a^{-31}+4845z^{17}a^{-33}-969z^{17}a^{-35}+153z^{17}a^{-37}-17z^{17}a^{-39}+z^{17}a^{-41}+100947z^{16}a^{-28}+80598z^{16}a^{-30}-15504z^{16}a^{-32}+3876z^{16}a^{-34}-816z^{16}a^{-36}+136z^{16}a^{-38}-16z^{16}a^{-40}+z^{16}a^{-42}+54264z^{15}a^{-29}+38760z^{15}a^{-31}-11628z^{15}a^{-33}+3060z^{15}a^{-35}-680z^{15}a^{-37}+120z^{15}a^{-39}-15z^{15}a^{-41}+z^{15}a^{-43}-170544z^{14}a^{-28}-131784z^{14}a^{-30}+27132z^{14}a^{-32}-8568z^{14}a^{-34}+2380z^{14}a^{-36}-560z^{14}a^{-38}+105z^{14}a^{-40}-14z^{14}a^{-42}+z^{14}a^{-44}-77520z^{13}a^{-29}-50388z^{13}a^{-31}+18564z^{13}a^{-33}-6188z^{13}a^{-35}+1820z^{13}a^{-37}-455z^{13}a^{-39}+91z^{13}a^{-41}-13z^{13}a^{-43}+z^{13}a^{-45}+203490z^{12}a^{-28}+153102z^{12}a^{-30}-31824z^{12}a^{-32}+12376z^{12}a^{-34}-4368z^{12}a^{-36}+1365z^{12}a^{-38}-364z^{12}a^{-40}+78z^{12}a^{-42}-12z^{12}a^{-44}+z^{12}a^{-46}+75582z^{11}a^{-29}+43758z^{11}a^{-31}-19448z^{11}a^{-33}+8008z^{11}a^{-35}-3003z^{11}a^{-37}+1001z^{11}a^{-39}-286z^{11}a^{-41}+66z^{11}a^{-43}-11z^{11}a^{-45}+z^{11}a^{-47}-167960z^{10}a^{-28}-124202z^{10}a^{-30}+24310z^{10}a^{-32}-11440z^{10}a^{-34}+5005z^{10}a^{-36}-2002z^{10}a^{-38}+715z^{10}a^{-40}-220z^{10}a^{-42}+55z^{10}a^{-44}-10z^{10}a^{-46}+z^{10}a^{-48}-48620z^9a^{-29}-24310z^9a^{-31}+12870z^9a^{-33}-6435z^9a^{-35}+3003z^9a^{-37}-1287z^9a^{-39}+495z^9a^{-41}-165z^9a^{-43}+45z^9a^{-45}-9z^9a^{-47}+z^9a^{-49}+92378z^8a^{-28}+68068z^8a^{-30}-11440z^8a^{-32}+6435z^8a^{-34}-3432z^8a^{-36}+1716z^8a^{-38}-792z^8a^{-40}+330z^8a^{-42}-120z^8a^{-44}+36z^8a^{-46}-8z^8a^{-48}+z^8a^{-50}+19448z^7a^{-29}+8008z^7a^{-31}-5005z^7a^{-33}+3003z^7a^{-35}-1716z^7a^{-37}+924z^7a^{-39}-462z^7a^{-41}+210z^7a^{-43}-84z^7a^{-45}+28z^7a^{-47}-7z^7a^{-49}+z^7a^{-51}-31824z^6a^{-28}-23816z^6a^{-30}+3003z^6a^{-32}-2002z^6a^{-34}+1287z^6a^{-36}-792z^6a^{-38}+462z^6a^{-40}-252z^6a^{-42}+126z^6a^{-44}-56z^6a^{-46}+21z^6a^{-48}-6z^6a^{-50}+z^6a^{-52}-4368z^5a^{-29}-1365z^5a^{-31}+1001z^5a^{-33}-715z^5a^{-35}+495z^5a^{-37}-330z^5a^{-39}+210z^5a^{-41}-126z^5a^{-43}+70z^5a^{-45}-35z^5a^{-47}+15z^5a^{-49}-5z^5a^{-51}+z^5a^{-53}+6188z^4a^{-28}+4823z^4a^{-30}-364z^4a^{-32}+286z^4a^{-34}-220z^4a^{-36}+165z^4a^{-38}-120z^4a^{-40}+84z^4a^{-42}-56z^4a^{-44}+35z^4a^{-46}-20z^4a^{-48}+10z^4a^{-50}-4z^4a^{-52}+z^4a^{-54}+455z^3a^{-29}+91z^3a^{-31}-78z^3a^{-33}+66z^3a^{-35}-55z^3a^{-37}+45z^3a^{-39}-36z^3a^{-41}+28z^3a^{-43}-21z^3a^{-45}+15z^3a^{-47}-10z^3a^{-49}+6z^3a^{-51}-3z^3a^{-53}+z^3a^{-55}-560z^2a^{-28}-469z^2a^{-30}+13z^2a^{-32}-12z^2a^{-34}+11z^2a^{-36}-10z^2a^{-38}+9z^2a^{-40}-8z^2a^{-42}+7z^2a^{-44}-6z^2a^{-46}+5z^2a^{-48}-4z^2a^{-50}+3z^2a^{-52}-2z^2a^{-54}+z^2a^{-56}-14za^{-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}-za^{-55}+za^{-57}+15a^{-28}+14a^{-30} }[/math]

Vassiliev invariants

V2 and V3:

(105, 1015)

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(29,2)/V 2,1 Data:T(29,2)/V 3,1 Data:T(29,2)/V 4,1 Data:T(29,2)/V 4,2 Data:T(29,2)/V 4,3 Data:T(29,2)/V 5,1 Data:T(29,2)/V 5,2 Data:T(29,2)/V 5,3 Data:T(29,2)/V 5,4 Data:T(29,2)/V 6,1 Data:T(29,2)/V 6,2 Data:T(29,2)/V 6,3 Data:T(29,2)/V 6,4 Data:T(29,2)/V 6,5 Data:T(29,2)/V 6,6 Data:T(29,2)/V 6,7 Data:T(29,2)/V 6,8 Data:T(29,2)/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]28 is the signature of T(29,2). 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 28 29 χ 87                                                           1 -1 85                                                             0 83                                                       1 1   0 81                                                             0 79                                                   1 1       0 77                                                             0 75                                               1 1           0 73                                                             0 71                                           1 1               0 69                                                             0 67                                       1 1                   0 65                                                             0 63                                   1 1                       0 61                                                             0 59                               1 1                           0 57                                                             0 55                           1 1                               0 53                                                             0 51                       1 1                                   0 49                                                             0 47                   1 1                                       0 45                                                             0 43               1 1                                           0 41                                                             0 39           1 1                                               0 37                                                             0 35       1 1                                                   0 33                                                             0 31     1                                                       1 29 1                                                           1 27 1                                                           1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=27 }[/math] [math]\displaystyle{ i=29 }[/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] [math]\displaystyle{ r=28 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=29 }[/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[29, 2]]
Out[2]=	29
In[3]:=	PD[TorusKnot[29, 2]]
Out[3]=	PD[X[19, 49, 20, 48], X[49, 21, 50, 20], X[21, 51, 22, 50], X[51, 23, 52, 22], X[23, 53, 24, 52], X[53, 25, 54, 24], X[25, 55, 26, 54], X[55, 27, 56, 26], X[27, 57, 28, 56], X[57, 29, 58, 28], X[29, 1, 30, 58], X[1, 31, 2, 30], X[31, 3, 32, 2], X[3, 33, 4, 32], X[33, 5, 34, 4], X[5, 35, 6, 34], X[35, 7, 36, 6], X[7, 37, 8, 36], X[37, 9, 38, 8], X[9, 39, 10, 38], X[39, 11, 40, 10], X[11, 41, 12, 40], X[41, 13, 42, 12], X[13, 43, 14, 42], X[43, 15, 44, 14], X[15, 45, 16, 44], X[45, 17, 46, 16], X[17, 47, 18, 46], X[47, 19, 48, 18]]
In[4]:=	GaussCode[TorusKnot[29, 2]]
Out[4]=	GaussCode[-12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -28, 29, -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, 28, -29, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11]
In[5]:=	BR[TorusKnot[29, 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, 1, 1}]
In[6]:=	alex = Alexander[TorusKnot[29, 2]][t]
Out[6]=	-14 -13 -12 -11 1 + Alternating - 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 14 Alternating + Alternating - Alternating + Alternating
In[7]:=	Conway[TorusKnot[29, 2]][z]
Out[7]=	2 4 6 8 10 12 1 + 105 z + 1820 z + 12376 z + 43758 z + 92378 z + 125970 z + 14 16 18 20 22 24 116280 z + 74613 z + 33649 z + 10626 z + 2300 z + 325 z + 26 28 27 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[TorusKnot[29, 2]], KnotSignature[TorusKnot[29, 2]]}
Out[9]=	{29, 28}
In[10]:=	J=Jones[TorusKnot[29, 2]][q]
Out[10]=	14 16 17 18 19 20 21 22 23 24 25 26 q + q - q + q - q + q - q + q - q + q - q + q - 27 28 29 30 31 32 33 34 35 36 37 q + q - q + q - q + q - q + q - q + q - q + 38 39 40 41 42 43 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[29, 2]][q]
Out[12]=	NotAvailable
In[13]:=	Kauffman[TorusKnot[29, 2]][a, z]
Out[13]=	NotAvailable
In[14]:=	{Vassiliev[2][TorusKnot[29, 2]], Vassiliev[3][TorusKnot[29, 2]]}
Out[14]=	{0, 1015}
In[15]:=	Kh[TorusKnot[29, 2]][q, t]
Out[15]=	27 29 2 31 3 35 4 35 q + q + Alternating q + Alternating q + Alternating q + 5 39 6 39 7 43 Alternating q + Alternating q + Alternating q + 8 43 9 47 10 47 Alternating q + Alternating q + Alternating q + 11 51 12 51 13 55 Alternating q + Alternating q + Alternating q + 14 55 15 59 16 59 Alternating q + Alternating q + Alternating q + 17 63 18 63 19 67 Alternating q + Alternating q + Alternating q + 20 67 21 71 22 71 Alternating q + Alternating q + Alternating q + 23 75 24 75 25 79 Alternating q + Alternating q + Alternating q + 26 79 27 83 28 83 Alternating q + Alternating q + Alternating q + 29 87 Alternating q