T(31,2)

T(29,2)

T(8,5)

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	T(31,2) Quick Notes

T(31,2) Further Notes and Views

Knot presentations

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

Polynomial invariants

Alexander polynomial	$t^{15}-t^{14}+t^{13}-t^{12}+t^{11}-t^{10}+t^{9}-t^{8}+t^{7}-t^{6}+t^{5}-t^{4}+t^{3}-t^{2}+t-1+t^{-1}-t^{-2}+t^{-3}-t^{-4}+t^{-5}-t^{-6}+t^{-7}-t^{-8}+t^{-9}-t^{-10}+t^{-11}-t^{-12}+t^{-13}-t^{-14}+t^{-15}$
Conway polynomial	$z^{30}+29z^{28}+378z^{26}+2925z^{24}+14950z^{22}+53130z^{20}+134596z^{18}+245157z^{16}+319770z^{14}+293930z^{12}+184756z^{10}+75582z^{8}+18564z^{6}+2380z^{4}+120z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 31, 30 }
Jones polynomial	$-q^{46}+q^{45}-q^{44}+q^{43}-q^{42}+q^{41}-q^{40}+q^{39}-q^{38}+q^{37}-q^{36}+q^{35}-q^{34}+q^{33}-q^{32}+q^{31}-q^{30}+q^{29}-q^{28}+q^{27}-q^{26}+q^{25}-q^{24}+q^{23}-q^{22}+q^{21}-q^{20}+q^{19}-q^{18}+q^{17}+q^{15}$
HOMFLY-PT polynomial (db, data sources)	$z^{30}a^{-30}-30z^{28}a^{-30}-z^{28}a^{-32}+406z^{26}a^{-30}+28z^{26}a^{-32}-3276z^{24}a^{-30}-351z^{24}a^{-32}+17550z^{22}a^{-30}+2600z^{22}a^{-32}-65780z^{20}a^{-30}-12650z^{20}a^{-32}+177100z^{18}a^{-30}+42504z^{18}a^{-32}-346104z^{16}a^{-30}-100947z^{16}a^{-32}+490314z^{14}a^{-30}+170544z^{14}a^{-32}-497420z^{12}a^{-30}-203490z^{12}a^{-32}+352716z^{10}a^{-30}+167960z^{10}a^{-32}-167960z^{8}a^{-30}-92378z^{8}a^{-32}+50388z^{6}a^{-30}+31824z^{6}a^{-32}-8568z^{4}a^{-30}-6188z^{4}a^{-32}+680z^{2}a^{-30}+560z^{2}a^{-32}-16a^{-30}-15a^{-32}$
Kauffman polynomial (db, data sources)	$z^{30}a^{-30}+z^{30}a^{-32}+z^{29}a^{-31}+z^{29}a^{-33}-30z^{28}a^{-30}-29z^{28}a^{-32}+z^{28}a^{-34}-28z^{27}a^{-31}-27z^{27}a^{-33}+z^{27}a^{-35}+406z^{26}a^{-30}+379z^{26}a^{-32}-26z^{26}a^{-34}+z^{26}a^{-36}+351z^{25}a^{-31}+325z^{25}a^{-33}-25z^{25}a^{-35}+z^{25}a^{-37}-3276z^{24}a^{-30}-2951z^{24}a^{-32}+300z^{24}a^{-34}-24z^{24}a^{-36}+z^{24}a^{-38}-2600z^{23}a^{-31}-2300z^{23}a^{-33}+276z^{23}a^{-35}-23z^{23}a^{-37}+z^{23}a^{-39}+17550z^{22}a^{-30}+15250z^{22}a^{-32}-2024z^{22}a^{-34}+253z^{22}a^{-36}-22z^{22}a^{-38}+z^{22}a^{-40}+12650z^{21}a^{-31}+10626z^{21}a^{-33}-1771z^{21}a^{-35}+231z^{21}a^{-37}-21z^{21}a^{-39}+z^{21}a^{-41}-65780z^{20}a^{-30}-55154z^{20}a^{-32}+8855z^{20}a^{-34}-1540z^{20}a^{-36}+210z^{20}a^{-38}-20z^{20}a^{-40}+z^{20}a^{-42}-42504z^{19}a^{-31}-33649z^{19}a^{-33}+7315z^{19}a^{-35}-1330z^{19}a^{-37}+190z^{19}a^{-39}-19z^{19}a^{-41}+z^{19}a^{-43}+177100z^{18}a^{-30}+143451z^{18}a^{-32}-26334z^{18}a^{-34}+5985z^{18}a^{-36}-1140z^{18}a^{-38}+171z^{18}a^{-40}-18z^{18}a^{-42}+z^{18}a^{-44}+100947z^{17}a^{-31}+74613z^{17}a^{-33}-20349z^{17}a^{-35}+4845z^{17}a^{-37}-969z^{17}a^{-39}+153z^{17}a^{-41}-17z^{17}a^{-43}+z^{17}a^{-45}-346104z^{16}a^{-30}-271491z^{16}a^{-32}+54264z^{16}a^{-34}-15504z^{16}a^{-36}+3876z^{16}a^{-38}-816z^{16}a^{-40}+136z^{16}a^{-42}-16z^{16}a^{-44}+z^{16}a^{-46}-170544z^{15}a^{-31}-116280z^{15}a^{-33}+38760z^{15}a^{-35}-11628z^{15}a^{-37}+3060z^{15}a^{-39}-680z^{15}a^{-41}+120z^{15}a^{-43}-15z^{15}a^{-45}+z^{15}a^{-47}+490314z^{14}a^{-30}+374034z^{14}a^{-32}-77520z^{14}a^{-34}+27132z^{14}a^{-36}-8568z^{14}a^{-38}+2380z^{14}a^{-40}-560z^{14}a^{-42}+105z^{14}a^{-44}-14z^{14}a^{-46}+z^{14}a^{-48}+203490z^{13}a^{-31}+125970z^{13}a^{-33}-50388z^{13}a^{-35}+18564z^{13}a^{-37}-6188z^{13}a^{-39}+1820z^{13}a^{-41}-455z^{13}a^{-43}+91z^{13}a^{-45}-13z^{13}a^{-47}+z^{13}a^{-49}-497420z^{12}a^{-30}-371450z^{12}a^{-32}+75582z^{12}a^{-34}-31824z^{12}a^{-36}+12376z^{12}a^{-38}-4368z^{12}a^{-40}+1365z^{12}a^{-42}-364z^{12}a^{-44}+78z^{12}a^{-46}-12z^{12}a^{-48}+z^{12}a^{-50}-167960z^{11}a^{-31}-92378z^{11}a^{-33}+43758z^{11}a^{-35}-19448z^{11}a^{-37}+8008z^{11}a^{-39}-3003z^{11}a^{-41}+1001z^{11}a^{-43}-286z^{11}a^{-45}+66z^{11}a^{-47}-11z^{11}a^{-49}+z^{11}a^{-51}+352716z^{10}a^{-30}+260338z^{10}a^{-32}-48620z^{10}a^{-34}+24310z^{10}a^{-36}-11440z^{10}a^{-38}+5005z^{10}a^{-40}-2002z^{10}a^{-42}+715z^{10}a^{-44}-220z^{10}a^{-46}+55z^{10}a^{-48}-10z^{10}a^{-50}+z^{10}a^{-52}+92378z^{9}a^{-31}+43758z^{9}a^{-33}-24310z^{9}a^{-35}+12870z^{9}a^{-37}-6435z^{9}a^{-39}+3003z^{9}a^{-41}-1287z^{9}a^{-43}+495z^{9}a^{-45}-165z^{9}a^{-47}+45z^{9}a^{-49}-9z^{9}a^{-51}+z^{9}a^{-53}-167960z^{8}a^{-30}-124202z^{8}a^{-32}+19448z^{8}a^{-34}-11440z^{8}a^{-36}+6435z^{8}a^{-38}-3432z^{8}a^{-40}+1716z^{8}a^{-42}-792z^{8}a^{-44}+330z^{8}a^{-46}-120z^{8}a^{-48}+36z^{8}a^{-50}-8z^{8}a^{-52}+z^{8}a^{-54}-31824z^{7}a^{-31}-12376z^{7}a^{-33}+8008z^{7}a^{-35}-5005z^{7}a^{-37}+3003z^{7}a^{-39}-1716z^{7}a^{-41}+924z^{7}a^{-43}-462z^{7}a^{-45}+210z^{7}a^{-47}-84z^{7}a^{-49}+28z^{7}a^{-51}-7z^{7}a^{-53}+z^{7}a^{-55}+50388z^{6}a^{-30}+38012z^{6}a^{-32}-4368z^{6}a^{-34}+3003z^{6}a^{-36}-2002z^{6}a^{-38}+1287z^{6}a^{-40}-792z^{6}a^{-42}+462z^{6}a^{-44}-252z^{6}a^{-46}+126z^{6}a^{-48}-56z^{6}a^{-50}+21z^{6}a^{-52}-6z^{6}a^{-54}+z^{6}a^{-56}+6188z^{5}a^{-31}+1820z^{5}a^{-33}-1365z^{5}a^{-35}+1001z^{5}a^{-37}-715z^{5}a^{-39}+495z^{5}a^{-41}-330z^{5}a^{-43}+210z^{5}a^{-45}-126z^{5}a^{-47}+70z^{5}a^{-49}-35z^{5}a^{-51}+15z^{5}a^{-53}-5z^{5}a^{-55}+z^{5}a^{-57}-8568z^{4}a^{-30}-6748z^{4}a^{-32}+455z^{4}a^{-34}-364z^{4}a^{-36}+286z^{4}a^{-38}-220z^{4}a^{-40}+165z^{4}a^{-42}-120z^{4}a^{-44}+84z^{4}a^{-46}-56z^{4}a^{-48}+35z^{4}a^{-50}-20z^{4}a^{-52}+10z^{4}a^{-54}-4z^{4}a^{-56}+z^{4}a^{-58}-560z^{3}a^{-31}-105z^{3}a^{-33}+91z^{3}a^{-35}-78z^{3}a^{-37}+66z^{3}a^{-39}-55z^{3}a^{-41}+45z^{3}a^{-43}-36z^{3}a^{-45}+28z^{3}a^{-47}-21z^{3}a^{-49}+15z^{3}a^{-51}-10z^{3}a^{-53}+6z^{3}a^{-55}-3z^{3}a^{-57}+z^{3}a^{-59}+680z^{2}a^{-30}+575z^{2}a^{-32}-14z^{2}a^{-34}+13z^{2}a^{-36}-12z^{2}a^{-38}+11z^{2}a^{-40}-10z^{2}a^{-42}+9z^{2}a^{-44}-8z^{2}a^{-46}+7z^{2}a^{-48}-6z^{2}a^{-50}+5z^{2}a^{-52}-4z^{2}a^{-54}+3z^{2}a^{-56}-2z^{2}a^{-58}+z^{2}a^{-60}+15za^{-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}-za^{-59}+za^{-61}-16a^{-30}-15a^{-32}$
The A2 invariant	Data:T(31,2)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(31,2)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(31,2) 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(31,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{15}-t^{14}+t^{13}-t^{12}+t^{11}-t^{10}+t^{9}-t^{8}+t^{7}-t^{6}+t^{5}-t^{4}+t^{3}-t^{2}+t-1+t^{-1}-t^{-2}+t^{-3}-t^{-4}+t^{-5}-t^{-6}+t^{-7}-t^{-8}+t^{-9}-t^{-10}+t^{-11}-t^{-12}+t^{-13}-t^{-14}+t^{-15}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{30}+29z^{28}+378z^{26}+2925z^{24}+14950z^{22}+53130z^{20}+134596z^{18}+245157z^{16}+319770z^{14}+293930z^{12}+184756z^{10}+75582z^{8}+18564z^{6}+2380z^{4}+120z^{2}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 31, 30 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{46}+q^{45}-q^{44}+q^{43}-q^{42}+q^{41}-q^{40}+q^{39}-q^{38}+q^{37}-q^{36}+q^{35}-q^{34}+q^{33}-q^{32}+q^{31}-q^{30}+q^{29}-q^{28}+q^{27}-q^{26}+q^{25}-q^{24}+q^{23}-q^{22}+q^{21}-q^{20}+q^{19}-q^{18}+q^{17}+q^{15}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{30}a^{-30}-30z^{28}a^{-30}-z^{28}a^{-32}+406z^{26}a^{-30}+28z^{26}a^{-32}-3276z^{24}a^{-30}-351z^{24}a^{-32}+17550z^{22}a^{-30}+2600z^{22}a^{-32}-65780z^{20}a^{-30}-12650z^{20}a^{-32}+177100z^{18}a^{-30}+42504z^{18}a^{-32}-346104z^{16}a^{-30}-100947z^{16}a^{-32}+490314z^{14}a^{-30}+170544z^{14}a^{-32}-497420z^{12}a^{-30}-203490z^{12}a^{-32}+352716z^{10}a^{-30}+167960z^{10}a^{-32}-167960z^{8}a^{-30}-92378z^{8}a^{-32}+50388z^{6}a^{-30}+31824z^{6}a^{-32}-8568z^{4}a^{-30}-6188z^{4}a^{-32}+680z^{2}a^{-30}+560z^{2}a^{-32}-16a^{-30}-15a^{-32}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{30}a^{-30}+z^{30}a^{-32}+z^{29}a^{-31}+z^{29}a^{-33}-30z^{28}a^{-30}-29z^{28}a^{-32}+z^{28}a^{-34}-28z^{27}a^{-31}-27z^{27}a^{-33}+z^{27}a^{-35}+406z^{26}a^{-30}+379z^{26}a^{-32}-26z^{26}a^{-34}+z^{26}a^{-36}+351z^{25}a^{-31}+325z^{25}a^{-33}-25z^{25}a^{-35}+z^{25}a^{-37}-3276z^{24}a^{-30}-2951z^{24}a^{-32}+300z^{24}a^{-34}-24z^{24}a^{-36}+z^{24}a^{-38}-2600z^{23}a^{-31}-2300z^{23}a^{-33}+276z^{23}a^{-35}-23z^{23}a^{-37}+z^{23}a^{-39}+17550z^{22}a^{-30}+15250z^{22}a^{-32}-2024z^{22}a^{-34}+253z^{22}a^{-36}-22z^{22}a^{-38}+z^{22}a^{-40}+12650z^{21}a^{-31}+10626z^{21}a^{-33}-1771z^{21}a^{-35}+231z^{21}a^{-37}-21z^{21}a^{-39}+z^{21}a^{-41}-65780z^{20}a^{-30}-55154z^{20}a^{-32}+8855z^{20}a^{-34}-1540z^{20}a^{-36}+210z^{20}a^{-38}-20z^{20}a^{-40}+z^{20}a^{-42}-42504z^{19}a^{-31}-33649z^{19}a^{-33}+7315z^{19}a^{-35}-1330z^{19}a^{-37}+190z^{19}a^{-39}-19z^{19}a^{-41}+z^{19}a^{-43}+177100z^{18}a^{-30}+143451z^{18}a^{-32}-26334z^{18}a^{-34}+5985z^{18}a^{-36}-1140z^{18}a^{-38}+171z^{18}a^{-40}-18z^{18}a^{-42}+z^{18}a^{-44}+100947z^{17}a^{-31}+74613z^{17}a^{-33}-20349z^{17}a^{-35}+4845z^{17}a^{-37}-969z^{17}a^{-39}+153z^{17}a^{-41}-17z^{17}a^{-43}+z^{17}a^{-45}-346104z^{16}a^{-30}-271491z^{16}a^{-32}+54264z^{16}a^{-34}-15504z^{16}a^{-36}+3876z^{16}a^{-38}-816z^{16}a^{-40}+136z^{16}a^{-42}-16z^{16}a^{-44}+z^{16}a^{-46}-170544z^{15}a^{-31}-116280z^{15}a^{-33}+38760z^{15}a^{-35}-11628z^{15}a^{-37}+3060z^{15}a^{-39}-680z^{15}a^{-41}+120z^{15}a^{-43}-15z^{15}a^{-45}+z^{15}a^{-47}+490314z^{14}a^{-30}+374034z^{14}a^{-32}-77520z^{14}a^{-34}+27132z^{14}a^{-36}-8568z^{14}a^{-38}+2380z^{14}a^{-40}-560z^{14}a^{-42}+105z^{14}a^{-44}-14z^{14}a^{-46}+z^{14}a^{-48}+203490z^{13}a^{-31}+125970z^{13}a^{-33}-50388z^{13}a^{-35}+18564z^{13}a^{-37}-6188z^{13}a^{-39}+1820z^{13}a^{-41}-455z^{13}a^{-43}+91z^{13}a^{-45}-13z^{13}a^{-47}+z^{13}a^{-49}-497420z^{12}a^{-30}-371450z^{12}a^{-32}+75582z^{12}a^{-34}-31824z^{12}a^{-36}+12376z^{12}a^{-38}-4368z^{12}a^{-40}+1365z^{12}a^{-42}-364z^{12}a^{-44}+78z^{12}a^{-46}-12z^{12}a^{-48}+z^{12}a^{-50}-167960z^{11}a^{-31}-92378z^{11}a^{-33}+43758z^{11}a^{-35}-19448z^{11}a^{-37}+8008z^{11}a^{-39}-3003z^{11}a^{-41}+1001z^{11}a^{-43}-286z^{11}a^{-45}+66z^{11}a^{-47}-11z^{11}a^{-49}+z^{11}a^{-51}+352716z^{10}a^{-30}+260338z^{10}a^{-32}-48620z^{10}a^{-34}+24310z^{10}a^{-36}-11440z^{10}a^{-38}+5005z^{10}a^{-40}-2002z^{10}a^{-42}+715z^{10}a^{-44}-220z^{10}a^{-46}+55z^{10}a^{-48}-10z^{10}a^{-50}+z^{10}a^{-52}+92378z^{9}a^{-31}+43758z^{9}a^{-33}-24310z^{9}a^{-35}+12870z^{9}a^{-37}-6435z^{9}a^{-39}+3003z^{9}a^{-41}-1287z^{9}a^{-43}+495z^{9}a^{-45}-165z^{9}a^{-47}+45z^{9}a^{-49}-9z^{9}a^{-51}+z^{9}a^{-53}-167960z^{8}a^{-30}-124202z^{8}a^{-32}+19448z^{8}a^{-34}-11440z^{8}a^{-36}+6435z^{8}a^{-38}-3432z^{8}a^{-40}+1716z^{8}a^{-42}-792z^{8}a^{-44}+330z^{8}a^{-46}-120z^{8}a^{-48}+36z^{8}a^{-50}-8z^{8}a^{-52}+z^{8}a^{-54}-31824z^{7}a^{-31}-12376z^{7}a^{-33}+8008z^{7}a^{-35}-5005z^{7}a^{-37}+3003z^{7}a^{-39}-1716z^{7}a^{-41}+924z^{7}a^{-43}-462z^{7}a^{-45}+210z^{7}a^{-47}-84z^{7}a^{-49}+28z^{7}a^{-51}-7z^{7}a^{-53}+z^{7}a^{-55}+50388z^{6}a^{-30}+38012z^{6}a^{-32}-4368z^{6}a^{-34}+3003z^{6}a^{-36}-2002z^{6}a^{-38}+1287z^{6}a^{-40}-792z^{6}a^{-42}+462z^{6}a^{-44}-252z^{6}a^{-46}+126z^{6}a^{-48}-56z^{6}a^{-50}+21z^{6}a^{-52}-6z^{6}a^{-54}+z^{6}a^{-56}+6188z^{5}a^{-31}+1820z^{5}a^{-33}-1365z^{5}a^{-35}+1001z^{5}a^{-37}-715z^{5}a^{-39}+495z^{5}a^{-41}-330z^{5}a^{-43}+210z^{5}a^{-45}-126z^{5}a^{-47}+70z^{5}a^{-49}-35z^{5}a^{-51}+15z^{5}a^{-53}-5z^{5}a^{-55}+z^{5}a^{-57}-8568z^{4}a^{-30}-6748z^{4}a^{-32}+455z^{4}a^{-34}-364z^{4}a^{-36}+286z^{4}a^{-38}-220z^{4}a^{-40}+165z^{4}a^{-42}-120z^{4}a^{-44}+84z^{4}a^{-46}-56z^{4}a^{-48}+35z^{4}a^{-50}-20z^{4}a^{-52}+10z^{4}a^{-54}-4z^{4}a^{-56}+z^{4}a^{-58}-560z^{3}a^{-31}-105z^{3}a^{-33}+91z^{3}a^{-35}-78z^{3}a^{-37}+66z^{3}a^{-39}-55z^{3}a^{-41}+45z^{3}a^{-43}-36z^{3}a^{-45}+28z^{3}a^{-47}-21z^{3}a^{-49}+15z^{3}a^{-51}-10z^{3}a^{-53}+6z^{3}a^{-55}-3z^{3}a^{-57}+z^{3}a^{-59}+680z^{2}a^{-30}+575z^{2}a^{-32}-14z^{2}a^{-34}+13z^{2}a^{-36}-12z^{2}a^{-38}+11z^{2}a^{-40}-10z^{2}a^{-42}+9z^{2}a^{-44}-8z^{2}a^{-46}+7z^{2}a^{-48}-6z^{2}a^{-50}+5z^{2}a^{-52}-4z^{2}a^{-54}+3z^{2}a^{-56}-2z^{2}a^{-58}+z^{2}a^{-60}+15za^{-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}-za^{-59}+za^{-61}-16a^{-30}-15a^{-32}$

Vassiliev invariants

V2 and V3:

(120, 1240)

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(31,2)/V 2,1 Data:T(31,2)/V 3,1 Data:T(31,2)/V 4,1 Data:T(31,2)/V 4,2 Data:T(31,2)/V 4,3 Data:T(31,2)/V 5,1 Data:T(31,2)/V 5,2 Data:T(31,2)/V 5,3 Data:T(31,2)/V 5,4 Data:T(31,2)/V 6,1 Data:T(31,2)/V 6,2 Data:T(31,2)/V 6,3 Data:T(31,2)/V 6,4 Data:T(31,2)/V 6,5 Data:T(31,2)/V 6,6 Data:T(31,2)/V 6,7 Data:T(31,2)/V 6,8 Data:T(31,2)/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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$30 is the signature of T(31,2). 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 χ 93                                                               1 -1 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                                                       0 35                                                                 0 33     1                                                           1 31 1                                                               1 29 1                                                               1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=29$ $i=31$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=1$ $r=2$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=6$ ${\mathbb {Z} }$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=8$ ${\mathbb {Z} }$ $r=9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=10$ ${\mathbb {Z} }$ $r=11$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=12$ ${\mathbb {Z} }$ $r=13$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=14$ ${\mathbb {Z} }$ $r=15$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=16$ ${\mathbb {Z} }$ $r=17$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=18$ ${\mathbb {Z} }$ $r=19$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=20$ ${\mathbb {Z} }$ $r=21$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=22$ ${\mathbb {Z} }$ $r=23$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=24$ ${\mathbb {Z} }$ $r=25$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=26$ ${\mathbb {Z} }$ $r=27$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=28$ ${\mathbb {Z} }$ $r=29$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=30$ ${\mathbb {Z} }$ $r=31$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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[31, 2]]
Out[2]=	31
In[3]:=	PD[TorusKnot[31, 2]]
Out[3]=	PD[X[17, 49, 18, 48], X[49, 19, 50, 18], X[19, 51, 20, 50], X[51, 21, 52, 20], X[21, 53, 22, 52], X[53, 23, 54, 22], X[23, 55, 24, 54], X[55, 25, 56, 24], X[25, 57, 26, 56], X[57, 27, 58, 26], X[27, 59, 28, 58], X[59, 29, 60, 28], X[29, 61, 30, 60], X[61, 31, 62, 30], X[31, 1, 32, 62], X[1, 33, 2, 32], X[33, 3, 34, 2], X[3, 35, 4, 34], X[35, 5, 36, 4], X[5, 37, 6, 36], X[37, 7, 38, 6], X[7, 39, 8, 38], X[39, 9, 40, 8], X[9, 41, 10, 40], X[41, 11, 42, 10], X[11, 43, 12, 42], X[43, 13, 44, 12], X[13, 45, 14, 44], X[45, 15, 46, 14], X[15, 47, 16, 46], X[47, 17, 48, 16]]
In[4]:=	GaussCode[TorusKnot[31, 2]]
Out[4]=	GaussCode[-16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -28, 29, -30, 31, -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, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15]
In[5]:=	BR[TorusKnot[31, 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}]
In[6]:=	alex = Alexander[TorusKnot[31, 2]][t]
Out[6]=	-15 -14 -13 -1 + Alternating - Alternating + Alternating - -12 -11 -10 -9 Alternating + Alternating - Alternating + Alternating - -8 -7 -6 -5 Alternating + Alternating - Alternating + Alternating - -4 -3 -2 1 Alternating + Alternating - Alternating + ----------- + Alternating 2 3 4 Alternating - Alternating + Alternating - Alternating + 5 6 7 8 Alternating - Alternating + Alternating - Alternating + 9 10 11 12 Alternating - Alternating + Alternating - Alternating + 13 14 15 Alternating - Alternating + Alternating
In[7]:=	Conway[TorusKnot[31, 2]][z]
Out[7]=	2 4 6 8 10 12 1 + 120 z + 2380 z + 18564 z + 75582 z + 184756 z + 293930 z + 14 16 18 20 22 319770 z + 245157 z + 134596 z + 53130 z + 14950 z + 24 26 28 30 2925 z + 378 z + 29 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{}
In[9]:=	{KnotDet[TorusKnot[31, 2]], KnotSignature[TorusKnot[31, 2]]}
Out[9]=	{31, 30}
In[10]:=	J=Jones[TorusKnot[31, 2]][q]
Out[10]=	15 17 18 19 20 21 22 23 24 25 26 27 q + q - q + q - q + q - q + q - q + q - q + q - 28 29 30 31 32 33 34 35 36 37 38 q + q - q + q - q + q - q + q - q + q - q + 39 40 41 42 43 44 45 46 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[31, 2]][q]
Out[12]=	NotAvailable
In[13]:=	Kauffman[TorusKnot[31, 2]][a, z]
Out[13]=	NotAvailable
In[14]:=	{Vassiliev[2][TorusKnot[31, 2]], Vassiliev[3][TorusKnot[31, 2]]}
Out[14]=	{0, 1240}
In[15]:=	Kh[TorusKnot[31, 2]][q, t]
Out[15]=	29 31 2 33 3 37 4 37 q + q + Alternating q + Alternating q + Alternating q + 5 41 6 41 7 45 Alternating q + Alternating q + Alternating q + 8 45 9 49 10 49 Alternating q + Alternating q + Alternating q + 11 53 12 53 13 57 Alternating q + Alternating q + Alternating q + 14 57 15 61 16 61 Alternating q + Alternating q + Alternating q + 17 65 18 65 19 69 Alternating q + Alternating q + Alternating q + 20 69 21 73 22 73 Alternating q + Alternating q + Alternating q + 23 77 24 77 25 81 Alternating q + Alternating q + Alternating q + 26 81 27 85 28 85 Alternating q + Alternating q + Alternating q + 29 89 30 89 31 93 Alternating q + Alternating q + Alternating q