T(25,2)

T(25,2) Quantum Invariants.

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(25,2)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t12−t11 + t10−t9 + t8−t7 + t6−t5 + t4−t3 + t2−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

In[5]:=

Conway[K][z]

Out[5]=

z24 + 23z22 + 231z20 + 1330z18 + 4845z16 + 11628z14 + 18564z12 + 19448z10 + 12870z8 + 5005z6 + 1001z4 + 78z2 + 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]=

{ 25, 24 }

In[8]:=

Jones[K][q]

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

Out[8]=

−q37 + q36−q35 + q34−q33 + q32−q31 + q30−q29 + q28−q27 + q26−q25 + q24−q23 + q22−q21 + q20−q19 + q18−q17 + q16−q15 + q14 + q12

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z24a−24−24z22a−24−z22a−26 + 253z20a−24 + 22z20a−26−1540z18a−24−210z18a−26 + 5985z16a−24 + 1140z16a−26−15504z14a−24−3876z14a−26 + 27132z12a−24 + 8568z12a−26−31824z10a−24−12376z10a−26 + 24310z8a−24 + 11440z8a−26−11440z6a−24−6435z6a−26 + 3003z4a−24 + 2002z4a−26−364z2a−24−286z2a−26 + 13a−24 + 12a−26

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z24a−24 + z24a−26 + z23a−25 + z23a−27−24z22a−24−23z22a−26 + z22a−28−22z21a−25−21z21a−27 + z21a−29 + 253z20a−24 + 232z20a−26−20z20a−28 + z20a−30 + 210z19a−25 + 190z19a−27−19z19a−29 + z19a−31−1540z18a−24−1350z18a−26 + 171z18a−28−18z18a−30 + z18a−32−1140z17a−25−969z17a−27 + 153z17a−29−17z17a−31 + z17a−33 + 5985z16a−24 + 5016z16a−26−816z16a−28 + 136z16a−30−16z16a−32 + z16a−34 + 3876z15a−25 + 3060z15a−27−680z15a−29 + 120z15a−31−15z15a−33 + z15a−35−15504z14a−24−12444z14a−26 + 2380z14a−28−560z14a−30 + 105z14a−32−14z14a−34 + z14a−36−8568z13a−25−6188z13a−27 + 1820z13a−29−455z13a−31 + 91z13a−33−13z13a−35 + z13a−37 + 27132z12a−24 + 20944z12a−26−4368z12a−28 + 1365z12a−30−364z12a−32 + 78z12a−34−12z12a−36 + z12a−38 + 12376z11a−25 + 8008z11a−27−3003z11a−29 + 1001z11a−31−286z11a−33 + 66z11a−35−11z11a−37 + z11a−39−31824z10a−24−23816z10a−26 + 5005z10a−28−2002z10a−30 + 715z10a−32−220z10a−34 + 55z10a−36−10z10a−38 + z10a−40−11440z9a−25−6435z9a−27 + 3003z9a−29−1287z9a−31 + 495z9a−33−165z9a−35 + 45z9a−37−9z9a−39 + z9a−41 + 24310z8a−24 + 17875z8a−26−3432z8a−28 + 1716z8a−30−792z8a−32 + 330z8a−34−120z8a−36 + 36z8a−38−8z8a−40 + z8a−42 + 6435z7a−25 + 3003z7a−27−1716z7a−29 + 924z7a−31−462z7a−33 + 210z7a−35−84z7a−37 + 28z7a−39−7z7a−41 + z7a−43−11440z6a−24−8437z6a−26 + 1287z6a−28−792z6a−30 + 462z6a−32−252z6a−34 + 126z6a−36−56z6a−38 + 21z6a−40−6z6a−42 + z6a−44−2002z5a−25−715z5a−27 + 495z5a−29−330z5a−31 + 210z5a−33−126z5a−35 + 70z5a−37−35z5a−39 + 15z5a−41−5z5a−43 + z5a−45 + 3003z4a−24 + 2288z4a−26−220z4a−28 + 165z4a−30−120z4a−32 + 84z4a−34−56z4a−36 + 35z4a−38−20z4a−40 + 10z4a−42−4z4a−44 + z4a−46 + 286z3a−25 + 66z3a−27−55z3a−29 + 45z3a−31−36z3a−33 + 28z3a−35−21z3a−37 + 15z3a−39−10z3a−41 + 6z3a−43−3z3a−45 + z3a−47−364z2a−24−298z2a−26 + 11z2a−28−10z2a−30 + 9z2a−32−8z2a−34 + 7z2a−36−6z2a−38 + 5z2a−40−4z2a−42 + 3z2a−44−2z2a−46 + z2a−48−12za−25−za−27 + za−29−za−31 + za−33−za−35 + za−37−za−39 + za−41−za−43 + za−45−za−47 + za−49 + 13a−24 + 12a−26

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

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["T(25,2)"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

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

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

Out[4]=

{ t12−t11 + t10−t9 + t8−t7 + t6−t5 + t4−t3 + t2−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, −q37 + q36−q35 + q34−q33 + q32−q31 + q30−q29 + q28−q27 + q26−q25 + q24−q23 + q22−q21 + q20−q19 + q18−q17 + q16−q15 + q14 + q12 }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}