T(35,2)

T(7,6)

T(9,5)

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	T(35,2) Quick Notes

T(35,2) Further Notes and Views

Knot presentations

Planar diagram presentation	X29,65,30,64 X65,31,66,30 X31,67,32,66 X67,33,68,32 X33,69,34,68 X69,35,70,34 X35,1,36,70 X1,37,2,36 X37,3,38,2 X3,39,4,38 X39,5,40,4 X5,41,6,40 X41,7,42,6 X7,43,8,42 X43,9,44,8 X9,45,10,44 X45,11,46,10 X11,47,12,46 X47,13,48,12 X13,49,14,48 X49,15,50,14 X15,51,16,50 X51,17,52,16 X17,53,18,52 X53,19,54,18 X19,55,20,54 X55,21,56,20 X21,57,22,56 X57,23,58,22 X23,59,24,58 X59,25,60,24 X25,61,26,60 X61,27,62,26 X27,63,28,62 X63,29,64,28
Gauss code	-8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -28, 29, -30, 31, -32, 33, -34, 35, -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, 30, -31, 32, -33, 34, -35, 1, -2, 3, -4, 5, -6, 7
Dowker-Thistlethwaite code	36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
Conway Notation	Data:T(35,2)/Conway Notation

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^{17}-t^{16}+t^{15}-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} + t^{-15} - t^{-16} + t^{-17} }[/math]
Conway polynomial	[math]\displaystyle{ z^{34}+33 z^{32}+496 z^{30}+4495 z^{28}+27405 z^{26}+118755 z^{24}+376740 z^{22}+888030 z^{20}+1562275 z^{18}+2042975 z^{16}+1961256 z^{14}+1352078 z^{12}+646646 z^{10}+203490 z^8+38760 z^6+3876 z^4+153 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 35, 34 }
Jones polynomial	[math]\displaystyle{ -q^{52}+q^{51}-q^{50}+q^{49}-q^{48}+q^{47}-q^{46}+q^{45}-q^{44}+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^{17} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^{34}a^{-34}-34z^{32}a^{-34}-z^{32}a^{-36}+528z^{30}a^{-34}+32z^{30}a^{-36}-4960z^{28}a^{-34}-465z^{28}a^{-36}+31465z^{26}a^{-34}+4060z^{26}a^{-36}-142506z^{24}a^{-34}-23751z^{24}a^{-36}+475020z^{22}a^{-34}+98280z^{22}a^{-36}-1184040z^{20}a^{-34}-296010z^{20}a^{-36}+2220075z^{18}a^{-34}+657800z^{18}a^{-36}-3124550z^{16}a^{-34}-1081575z^{16}a^{-36}+3268760z^{14}a^{-34}+1307504z^{14}a^{-36}-2496144z^{12}a^{-34}-1144066z^{12}a^{-36}+1352078z^{10}a^{-34}+705432z^{10}a^{-36}-497420z^8a^{-34}-293930z^8a^{-36}+116280z^6a^{-34}+77520z^6a^{-36}-15504z^4a^{-34}-11628z^4a^{-36}+969z^2a^{-34}+816z^2a^{-36}-18a^{-34}-17a^{-36} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{34}a^{-34}+z^{34}a^{-36}+z^{33}a^{-35}+z^{33}a^{-37}-34z^{32}a^{-34}-33z^{32}a^{-36}+z^{32}a^{-38}-32z^{31}a^{-35}-31z^{31}a^{-37}+z^{31}a^{-39}+528z^{30}a^{-34}+497z^{30}a^{-36}-30z^{30}a^{-38}+z^{30}a^{-40}+465z^{29}a^{-35}+435z^{29}a^{-37}-29z^{29}a^{-39}+z^{29}a^{-41}-4960z^{28}a^{-34}-4525z^{28}a^{-36}+406z^{28}a^{-38}-28z^{28}a^{-40}+z^{28}a^{-42}-4060z^{27}a^{-35}-3654z^{27}a^{-37}+378z^{27}a^{-39}-27z^{27}a^{-41}+z^{27}a^{-43}+31465z^{26}a^{-34}+27811z^{26}a^{-36}-3276z^{26}a^{-38}+351z^{26}a^{-40}-26z^{26}a^{-42}+z^{26}a^{-44}+23751z^{25}a^{-35}+20475z^{25}a^{-37}-2925z^{25}a^{-39}+325z^{25}a^{-41}-25z^{25}a^{-43}+z^{25}a^{-45}-142506z^{24}a^{-34}-122031z^{24}a^{-36}+17550z^{24}a^{-38}-2600z^{24}a^{-40}+300z^{24}a^{-42}-24z^{24}a^{-44}+z^{24}a^{-46}-98280z^{23}a^{-35}-80730z^{23}a^{-37}+14950z^{23}a^{-39}-2300z^{23}a^{-41}+276z^{23}a^{-43}-23z^{23}a^{-45}+z^{23}a^{-47}+475020z^{22}a^{-34}+394290z^{22}a^{-36}-65780z^{22}a^{-38}+12650z^{22}a^{-40}-2024z^{22}a^{-42}+253z^{22}a^{-44}-22z^{22}a^{-46}+z^{22}a^{-48}+296010z^{21}a^{-35}+230230z^{21}a^{-37}-53130z^{21}a^{-39}+10626z^{21}a^{-41}-1771z^{21}a^{-43}+231z^{21}a^{-45}-21z^{21}a^{-47}+z^{21}a^{-49}-1184040z^{20}a^{-34}-953810z^{20}a^{-36}+177100z^{20}a^{-38}-42504z^{20}a^{-40}+8855z^{20}a^{-42}-1540z^{20}a^{-44}+210z^{20}a^{-46}-20z^{20}a^{-48}+z^{20}a^{-50}-657800z^{19}a^{-35}-480700z^{19}a^{-37}+134596z^{19}a^{-39}-33649z^{19}a^{-41}+7315z^{19}a^{-43}-1330z^{19}a^{-45}+190z^{19}a^{-47}-19z^{19}a^{-49}+z^{19}a^{-51}+2220075z^{18}a^{-34}+1739375z^{18}a^{-36}-346104z^{18}a^{-38}+100947z^{18}a^{-40}-26334z^{18}a^{-42}+5985z^{18}a^{-44}-1140z^{18}a^{-46}+171z^{18}a^{-48}-18z^{18}a^{-50}+z^{18}a^{-52}+1081575z^{17}a^{-35}+735471z^{17}a^{-37}-245157z^{17}a^{-39}+74613z^{17}a^{-41}-20349z^{17}a^{-43}+4845z^{17}a^{-45}-969z^{17}a^{-47}+153z^{17}a^{-49}-17z^{17}a^{-51}+z^{17}a^{-53}-3124550z^{16}a^{-34}-2389079z^{16}a^{-36}+490314z^{16}a^{-38}-170544z^{16}a^{-40}+54264z^{16}a^{-42}-15504z^{16}a^{-44}+3876z^{16}a^{-46}-816z^{16}a^{-48}+136z^{16}a^{-50}-16z^{16}a^{-52}+z^{16}a^{-54}-1307504z^{15}a^{-35}-817190z^{15}a^{-37}+319770z^{15}a^{-39}-116280z^{15}a^{-41}+38760z^{15}a^{-43}-11628z^{15}a^{-45}+3060z^{15}a^{-47}-680z^{15}a^{-49}+120z^{15}a^{-51}-15z^{15}a^{-53}+z^{15}a^{-55}+3268760z^{14}a^{-34}+2451570z^{14}a^{-36}-497420z^{14}a^{-38}+203490z^{14}a^{-40}-77520z^{14}a^{-42}+27132z^{14}a^{-44}-8568z^{14}a^{-46}+2380z^{14}a^{-48}-560z^{14}a^{-50}+105z^{14}a^{-52}-14z^{14}a^{-54}+z^{14}a^{-56}+1144066z^{13}a^{-35}+646646z^{13}a^{-37}-293930z^{13}a^{-39}+125970z^{13}a^{-41}-50388z^{13}a^{-43}+18564z^{13}a^{-45}-6188z^{13}a^{-47}+1820z^{13}a^{-49}-455z^{13}a^{-51}+91z^{13}a^{-53}-13z^{13}a^{-55}+z^{13}a^{-57}-2496144z^{12}a^{-34}-1849498z^{12}a^{-36}+352716z^{12}a^{-38}-167960z^{12}a^{-40}+75582z^{12}a^{-42}-31824z^{12}a^{-44}+12376z^{12}a^{-46}-4368z^{12}a^{-48}+1365z^{12}a^{-50}-364z^{12}a^{-52}+78z^{12}a^{-54}-12z^{12}a^{-56}+z^{12}a^{-58}-705432z^{11}a^{-35}-352716z^{11}a^{-37}+184756z^{11}a^{-39}-92378z^{11}a^{-41}+43758z^{11}a^{-43}-19448z^{11}a^{-45}+8008z^{11}a^{-47}-3003z^{11}a^{-49}+1001z^{11}a^{-51}-286z^{11}a^{-53}+66z^{11}a^{-55}-11z^{11}a^{-57}+z^{11}a^{-59}+1352078z^{10}a^{-34}+999362z^{10}a^{-36}-167960z^{10}a^{-38}+92378z^{10}a^{-40}-48620z^{10}a^{-42}+24310z^{10}a^{-44}-11440z^{10}a^{-46}+5005z^{10}a^{-48}-2002z^{10}a^{-50}+715z^{10}a^{-52}-220z^{10}a^{-54}+55z^{10}a^{-56}-10z^{10}a^{-58}+z^{10}a^{-60}+293930z^9a^{-35}+125970z^9a^{-37}-75582z^9a^{-39}+43758z^9a^{-41}-24310z^9a^{-43}+12870z^9a^{-45}-6435z^9a^{-47}+3003z^9a^{-49}-1287z^9a^{-51}+495z^9a^{-53}-165z^9a^{-55}+45z^9a^{-57}-9z^9a^{-59}+z^9a^{-61}-497420z^8a^{-34}-371450z^8a^{-36}+50388z^8a^{-38}-31824z^8a^{-40}+19448z^8a^{-42}-11440z^8a^{-44}+6435z^8a^{-46}-3432z^8a^{-48}+1716z^8a^{-50}-792z^8a^{-52}+330z^8a^{-54}-120z^8a^{-56}+36z^8a^{-58}-8z^8a^{-60}+z^8a^{-62}-77520z^7a^{-35}-27132z^7a^{-37}+18564z^7a^{-39}-12376z^7a^{-41}+8008z^7a^{-43}-5005z^7a^{-45}+3003z^7a^{-47}-1716z^7a^{-49}+924z^7a^{-51}-462z^7a^{-53}+210z^7a^{-55}-84z^7a^{-57}+28z^7a^{-59}-7z^7a^{-61}+z^7a^{-63}+116280z^6a^{-34}+89148z^6a^{-36}-8568z^6a^{-38}+6188z^6a^{-40}-4368z^6a^{-42}+3003z^6a^{-44}-2002z^6a^{-46}+1287z^6a^{-48}-792z^6a^{-50}+462z^6a^{-52}-252z^6a^{-54}+126z^6a^{-56}-56z^6a^{-58}+21z^6a^{-60}-6z^6a^{-62}+z^6a^{-64}+11628z^5a^{-35}+3060z^5a^{-37}-2380z^5a^{-39}+1820z^5a^{-41}-1365z^5a^{-43}+1001z^5a^{-45}-715z^5a^{-47}+495z^5a^{-49}-330z^5a^{-51}+210z^5a^{-53}-126z^5a^{-55}+70z^5a^{-57}-35z^5a^{-59}+15z^5a^{-61}-5z^5a^{-63}+z^5a^{-65}-15504z^4a^{-34}-12444z^4a^{-36}+680z^4a^{-38}-560z^4a^{-40}+455z^4a^{-42}-364z^4a^{-44}+286z^4a^{-46}-220z^4a^{-48}+165z^4a^{-50}-120z^4a^{-52}+84z^4a^{-54}-56z^4a^{-56}+35z^4a^{-58}-20z^4a^{-60}+10z^4a^{-62}-4z^4a^{-64}+z^4a^{-66}-816z^3a^{-35}-136z^3a^{-37}+120z^3a^{-39}-105z^3a^{-41}+91z^3a^{-43}-78z^3a^{-45}+66z^3a^{-47}-55z^3a^{-49}+45z^3a^{-51}-36z^3a^{-53}+28z^3a^{-55}-21z^3a^{-57}+15z^3a^{-59}-10z^3a^{-61}+6z^3a^{-63}-3z^3a^{-65}+z^3a^{-67}+969z^2a^{-34}+833z^2a^{-36}-16z^2a^{-38}+15z^2a^{-40}-14z^2a^{-42}+13z^2a^{-44}-12z^2a^{-46}+11z^2a^{-48}-10z^2a^{-50}+9z^2a^{-52}-8z^2a^{-54}+7z^2a^{-56}-6z^2a^{-58}+5z^2a^{-60}-4z^2a^{-62}+3z^2a^{-64}-2z^2a^{-66}+z^2a^{-68}+17za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-za^{-47}+za^{-49}-za^{-51}+za^{-53}-za^{-55}+za^{-57}-za^{-59}+za^{-61}-za^{-63}+za^{-65}-za^{-67}+za^{-69}-18a^{-34}-17a^{-36} }[/math]
The A2 invariant	Data:T(35,2)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(35,2)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(35,2) 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(35,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^{17}-t^{16}+t^{15}-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} + t^{-15} - t^{-16} + t^{-17} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^{34}+33 z^{32}+496 z^{30}+4495 z^{28}+27405 z^{26}+118755 z^{24}+376740 z^{22}+888030 z^{20}+1562275 z^{18}+2042975 z^{16}+1961256 z^{14}+1352078 z^{12}+646646 z^{10}+203490 z^8+38760 z^6+3876 z^4+153 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]=

{ 35, 34 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^{52}+q^{51}-q^{50}+q^{49}-q^{48}+q^{47}-q^{46}+q^{45}-q^{44}+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^{17} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^{34}a^{-34}-34z^{32}a^{-34}-z^{32}a^{-36}+528z^{30}a^{-34}+32z^{30}a^{-36}-4960z^{28}a^{-34}-465z^{28}a^{-36}+31465z^{26}a^{-34}+4060z^{26}a^{-36}-142506z^{24}a^{-34}-23751z^{24}a^{-36}+475020z^{22}a^{-34}+98280z^{22}a^{-36}-1184040z^{20}a^{-34}-296010z^{20}a^{-36}+2220075z^{18}a^{-34}+657800z^{18}a^{-36}-3124550z^{16}a^{-34}-1081575z^{16}a^{-36}+3268760z^{14}a^{-34}+1307504z^{14}a^{-36}-2496144z^{12}a^{-34}-1144066z^{12}a^{-36}+1352078z^{10}a^{-34}+705432z^{10}a^{-36}-497420z^8a^{-34}-293930z^8a^{-36}+116280z^6a^{-34}+77520z^6a^{-36}-15504z^4a^{-34}-11628z^4a^{-36}+969z^2a^{-34}+816z^2a^{-36}-18a^{-34}-17a^{-36} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^{34}a^{-34}+z^{34}a^{-36}+z^{33}a^{-35}+z^{33}a^{-37}-34z^{32}a^{-34}-33z^{32}a^{-36}+z^{32}a^{-38}-32z^{31}a^{-35}-31z^{31}a^{-37}+z^{31}a^{-39}+528z^{30}a^{-34}+497z^{30}a^{-36}-30z^{30}a^{-38}+z^{30}a^{-40}+465z^{29}a^{-35}+435z^{29}a^{-37}-29z^{29}a^{-39}+z^{29}a^{-41}-4960z^{28}a^{-34}-4525z^{28}a^{-36}+406z^{28}a^{-38}-28z^{28}a^{-40}+z^{28}a^{-42}-4060z^{27}a^{-35}-3654z^{27}a^{-37}+378z^{27}a^{-39}-27z^{27}a^{-41}+z^{27}a^{-43}+31465z^{26}a^{-34}+27811z^{26}a^{-36}-3276z^{26}a^{-38}+351z^{26}a^{-40}-26z^{26}a^{-42}+z^{26}a^{-44}+23751z^{25}a^{-35}+20475z^{25}a^{-37}-2925z^{25}a^{-39}+325z^{25}a^{-41}-25z^{25}a^{-43}+z^{25}a^{-45}-142506z^{24}a^{-34}-122031z^{24}a^{-36}+17550z^{24}a^{-38}-2600z^{24}a^{-40}+300z^{24}a^{-42}-24z^{24}a^{-44}+z^{24}a^{-46}-98280z^{23}a^{-35}-80730z^{23}a^{-37}+14950z^{23}a^{-39}-2300z^{23}a^{-41}+276z^{23}a^{-43}-23z^{23}a^{-45}+z^{23}a^{-47}+475020z^{22}a^{-34}+394290z^{22}a^{-36}-65780z^{22}a^{-38}+12650z^{22}a^{-40}-2024z^{22}a^{-42}+253z^{22}a^{-44}-22z^{22}a^{-46}+z^{22}a^{-48}+296010z^{21}a^{-35}+230230z^{21}a^{-37}-53130z^{21}a^{-39}+10626z^{21}a^{-41}-1771z^{21}a^{-43}+231z^{21}a^{-45}-21z^{21}a^{-47}+z^{21}a^{-49}-1184040z^{20}a^{-34}-953810z^{20}a^{-36}+177100z^{20}a^{-38}-42504z^{20}a^{-40}+8855z^{20}a^{-42}-1540z^{20}a^{-44}+210z^{20}a^{-46}-20z^{20}a^{-48}+z^{20}a^{-50}-657800z^{19}a^{-35}-480700z^{19}a^{-37}+134596z^{19}a^{-39}-33649z^{19}a^{-41}+7315z^{19}a^{-43}-1330z^{19}a^{-45}+190z^{19}a^{-47}-19z^{19}a^{-49}+z^{19}a^{-51}+2220075z^{18}a^{-34}+1739375z^{18}a^{-36}-346104z^{18}a^{-38}+100947z^{18}a^{-40}-26334z^{18}a^{-42}+5985z^{18}a^{-44}-1140z^{18}a^{-46}+171z^{18}a^{-48}-18z^{18}a^{-50}+z^{18}a^{-52}+1081575z^{17}a^{-35}+735471z^{17}a^{-37}-245157z^{17}a^{-39}+74613z^{17}a^{-41}-20349z^{17}a^{-43}+4845z^{17}a^{-45}-969z^{17}a^{-47}+153z^{17}a^{-49}-17z^{17}a^{-51}+z^{17}a^{-53}-3124550z^{16}a^{-34}-2389079z^{16}a^{-36}+490314z^{16}a^{-38}-170544z^{16}a^{-40}+54264z^{16}a^{-42}-15504z^{16}a^{-44}+3876z^{16}a^{-46}-816z^{16}a^{-48}+136z^{16}a^{-50}-16z^{16}a^{-52}+z^{16}a^{-54}-1307504z^{15}a^{-35}-817190z^{15}a^{-37}+319770z^{15}a^{-39}-116280z^{15}a^{-41}+38760z^{15}a^{-43}-11628z^{15}a^{-45}+3060z^{15}a^{-47}-680z^{15}a^{-49}+120z^{15}a^{-51}-15z^{15}a^{-53}+z^{15}a^{-55}+3268760z^{14}a^{-34}+2451570z^{14}a^{-36}-497420z^{14}a^{-38}+203490z^{14}a^{-40}-77520z^{14}a^{-42}+27132z^{14}a^{-44}-8568z^{14}a^{-46}+2380z^{14}a^{-48}-560z^{14}a^{-50}+105z^{14}a^{-52}-14z^{14}a^{-54}+z^{14}a^{-56}+1144066z^{13}a^{-35}+646646z^{13}a^{-37}-293930z^{13}a^{-39}+125970z^{13}a^{-41}-50388z^{13}a^{-43}+18564z^{13}a^{-45}-6188z^{13}a^{-47}+1820z^{13}a^{-49}-455z^{13}a^{-51}+91z^{13}a^{-53}-13z^{13}a^{-55}+z^{13}a^{-57}-2496144z^{12}a^{-34}-1849498z^{12}a^{-36}+352716z^{12}a^{-38}-167960z^{12}a^{-40}+75582z^{12}a^{-42}-31824z^{12}a^{-44}+12376z^{12}a^{-46}-4368z^{12}a^{-48}+1365z^{12}a^{-50}-364z^{12}a^{-52}+78z^{12}a^{-54}-12z^{12}a^{-56}+z^{12}a^{-58}-705432z^{11}a^{-35}-352716z^{11}a^{-37}+184756z^{11}a^{-39}-92378z^{11}a^{-41}+43758z^{11}a^{-43}-19448z^{11}a^{-45}+8008z^{11}a^{-47}-3003z^{11}a^{-49}+1001z^{11}a^{-51}-286z^{11}a^{-53}+66z^{11}a^{-55}-11z^{11}a^{-57}+z^{11}a^{-59}+1352078z^{10}a^{-34}+999362z^{10}a^{-36}-167960z^{10}a^{-38}+92378z^{10}a^{-40}-48620z^{10}a^{-42}+24310z^{10}a^{-44}-11440z^{10}a^{-46}+5005z^{10}a^{-48}-2002z^{10}a^{-50}+715z^{10}a^{-52}-220z^{10}a^{-54}+55z^{10}a^{-56}-10z^{10}a^{-58}+z^{10}a^{-60}+293930z^9a^{-35}+125970z^9a^{-37}-75582z^9a^{-39}+43758z^9a^{-41}-24310z^9a^{-43}+12870z^9a^{-45}-6435z^9a^{-47}+3003z^9a^{-49}-1287z^9a^{-51}+495z^9a^{-53}-165z^9a^{-55}+45z^9a^{-57}-9z^9a^{-59}+z^9a^{-61}-497420z^8a^{-34}-371450z^8a^{-36}+50388z^8a^{-38}-31824z^8a^{-40}+19448z^8a^{-42}-11440z^8a^{-44}+6435z^8a^{-46}-3432z^8a^{-48}+1716z^8a^{-50}-792z^8a^{-52}+330z^8a^{-54}-120z^8a^{-56}+36z^8a^{-58}-8z^8a^{-60}+z^8a^{-62}-77520z^7a^{-35}-27132z^7a^{-37}+18564z^7a^{-39}-12376z^7a^{-41}+8008z^7a^{-43}-5005z^7a^{-45}+3003z^7a^{-47}-1716z^7a^{-49}+924z^7a^{-51}-462z^7a^{-53}+210z^7a^{-55}-84z^7a^{-57}+28z^7a^{-59}-7z^7a^{-61}+z^7a^{-63}+116280z^6a^{-34}+89148z^6a^{-36}-8568z^6a^{-38}+6188z^6a^{-40}-4368z^6a^{-42}+3003z^6a^{-44}-2002z^6a^{-46}+1287z^6a^{-48}-792z^6a^{-50}+462z^6a^{-52}-252z^6a^{-54}+126z^6a^{-56}-56z^6a^{-58}+21z^6a^{-60}-6z^6a^{-62}+z^6a^{-64}+11628z^5a^{-35}+3060z^5a^{-37}-2380z^5a^{-39}+1820z^5a^{-41}-1365z^5a^{-43}+1001z^5a^{-45}-715z^5a^{-47}+495z^5a^{-49}-330z^5a^{-51}+210z^5a^{-53}-126z^5a^{-55}+70z^5a^{-57}-35z^5a^{-59}+15z^5a^{-61}-5z^5a^{-63}+z^5a^{-65}-15504z^4a^{-34}-12444z^4a^{-36}+680z^4a^{-38}-560z^4a^{-40}+455z^4a^{-42}-364z^4a^{-44}+286z^4a^{-46}-220z^4a^{-48}+165z^4a^{-50}-120z^4a^{-52}+84z^4a^{-54}-56z^4a^{-56}+35z^4a^{-58}-20z^4a^{-60}+10z^4a^{-62}-4z^4a^{-64}+z^4a^{-66}-816z^3a^{-35}-136z^3a^{-37}+120z^3a^{-39}-105z^3a^{-41}+91z^3a^{-43}-78z^3a^{-45}+66z^3a^{-47}-55z^3a^{-49}+45z^3a^{-51}-36z^3a^{-53}+28z^3a^{-55}-21z^3a^{-57}+15z^3a^{-59}-10z^3a^{-61}+6z^3a^{-63}-3z^3a^{-65}+z^3a^{-67}+969z^2a^{-34}+833z^2a^{-36}-16z^2a^{-38}+15z^2a^{-40}-14z^2a^{-42}+13z^2a^{-44}-12z^2a^{-46}+11z^2a^{-48}-10z^2a^{-50}+9z^2a^{-52}-8z^2a^{-54}+7z^2a^{-56}-6z^2a^{-58}+5z^2a^{-60}-4z^2a^{-62}+3z^2a^{-64}-2z^2a^{-66}+z^2a^{-68}+17za^{-35}+za^{-37}-za^{-39}+za^{-41}-za^{-43}+za^{-45}-za^{-47}+za^{-49}-za^{-51}+za^{-53}-za^{-55}+za^{-57}-za^{-59}+za^{-61}-za^{-63}+za^{-65}-za^{-67}+za^{-69}-18a^{-34}-17a^{-36} }[/math]

Vassiliev invariants

V2 and V3:

(153, 1785)

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(35,2)/V 2,1 Data:T(35,2)/V 3,1 Data:T(35,2)/V 4,1 Data:T(35,2)/V 4,2 Data:T(35,2)/V 4,3 Data:T(35,2)/V 5,1 Data:T(35,2)/V 5,2 Data:T(35,2)/V 5,3 Data:T(35,2)/V 5,4 Data:T(35,2)/V 6,1 Data:T(35,2)/V 6,2 Data:T(35,2)/V 6,3 Data:T(35,2)/V 6,4 Data:T(35,2)/V 6,5 Data:T(35,2)/V 6,6 Data:T(35,2)/V 6,7 Data:T(35,2)/V 6,8 Data:T(35,2)/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]34 is the signature of T(35,2). 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 30 31 32 33 34 35 χ 105                                                                       1 -1 103                                                                         0 101                                                                   1 1   0 99                                                                         0 97                                                               1 1       0 95                                                                         0 93                                                           1 1           0 91                                                                         0 89                                                       1 1               0 87                                                                         0 85                                                   1 1                   0 83                                                                         0 81                                               1 1                       0 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 35 1                                                                       1 33 1                                                                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=33 }[/math] [math]\displaystyle{ i=35 }[/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] [math]\displaystyle{ r=30 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=31 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=32 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=33 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=34 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=35 }[/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[35, 2]]
Out[2]=	35
In[3]:=	PD[TorusKnot[35, 2]]
Out[3]=	PD[X[29, 65, 30, 64], X[65, 31, 66, 30], X[31, 67, 32, 66], X[67, 33, 68, 32], X[33, 69, 34, 68], X[69, 35, 70, 34], X[35, 1, 36, 70], X[1, 37, 2, 36], X[37, 3, 38, 2], X[3, 39, 4, 38], X[39, 5, 40, 4], X[5, 41, 6, 40], X[41, 7, 42, 6], X[7, 43, 8, 42], X[43, 9, 44, 8], X[9, 45, 10, 44], X[45, 11, 46, 10], X[11, 47, 12, 46], X[47, 13, 48, 12], X[13, 49, 14, 48], X[49, 15, 50, 14], X[15, 51, 16, 50], X[51, 17, 52, 16], X[17, 53, 18, 52], X[53, 19, 54, 18], X[19, 55, 20, 54], X[55, 21, 56, 20], X[21, 57, 22, 56], X[57, 23, 58, 22], X[23, 59, 24, 58], X[59, 25, 60, 24], X[25, 61, 26, 60], X[61, 27, 62, 26], X[27, 63, 28, 62], X[63, 29, 64, 28]]
In[4]:=	GaussCode[TorusKnot[35, 2]]
Out[4]=	GaussCode[-8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -28, 29, -30, 31, -32, 33, -34, 35, -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, 30, -31, 32, -33, 34, -35, 1, -2, 3, -4, 5, -6, 7]
In[5]:=	BR[TorusKnot[35, 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, 1, 1, 1, 1, 1, 1}]
In[6]:=	alex = Alexander[TorusKnot[35, 2]][t]
Out[6]=	-17 -16 -15 -1 + Alternating - Alternating + Alternating - -14 -13 -12 -11 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 + 15 16 17 Alternating - Alternating + Alternating
In[7]:=	Conway[TorusKnot[35, 2]][z]
Out[7]=	2 4 6 8 10 1 + 153 z + 3876 z + 38760 z + 203490 z + 646646 z + 12 14 16 18 20 1352078 z + 1961256 z + 2042975 z + 1562275 z + 888030 z + 22 24 26 28 30 32 376740 z + 118755 z + 27405 z + 4495 z + 496 z + 33 z + 34 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[TorusKnot[35, 2]], KnotSignature[TorusKnot[35, 2]]}
Out[9]=	{35, 34}
In[10]:=	J=Jones[TorusKnot[35, 2]][q]
Out[10]=	17 19 20 21 22 23 24 25 26 27 28 29 q + q - q + q - q + q - q + q - q + q - q + q - 30 31 32 33 34 35 36 37 38 39 40 q + q - q + q - q + q - q + q - q + q - q + 41 42 43 44 45 46 47 48 49 50 51 52 q - q + q - 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[35, 2]][q]
Out[12]=	NotAvailable
In[13]:=	Kauffman[TorusKnot[35, 2]][a, z]
Out[13]=	NotAvailable
In[14]:=	{Vassiliev[2][TorusKnot[35, 2]], Vassiliev[3][TorusKnot[35, 2]]}
Out[14]=	{0, 1785}
In[15]:=	Kh[TorusKnot[35, 2]][q, t]
Out[15]=	33 35 2 37 3 41 4 41 q + q + Alternating q + Alternating q + Alternating q + 5 45 6 45 7 49 Alternating q + Alternating q + Alternating q + 8 49 9 53 10 53 Alternating q + Alternating q + Alternating q + 11 57 12 57 13 61 Alternating q + Alternating q + Alternating q + 14 61 15 65 16 65 Alternating q + Alternating q + Alternating q + 17 69 18 69 19 73 Alternating q + Alternating q + Alternating q + 20 73 21 77 22 77 Alternating q + Alternating q + Alternating q + 23 81 24 81 25 85 Alternating q + Alternating q + Alternating q + 26 85 27 89 28 89 Alternating q + Alternating q + Alternating q + 29 93 30 93 31 97 Alternating q + Alternating q + Alternating q + 32 97 33 101 34 101 Alternating q + Alternating q + Alternating q + 35 105 Alternating q