Rolfsen Splice Base
From Knot Atlas
|
[[Image:Data:Rolfsen Splice Base/Previous Knot.gif|80px]] |
[[Image:Data:Rolfsen Splice Base/Next Knot.gif|80px]] |
|
| Image:Rolfsen Splice Base.gif (KnotPlot image) |
See the full Rolfsen Knot Table. Visit <*n*>&id=<*k*> Rolfsen Splice Base's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit Data:Rolfsen Splice Base/KnotilusURL at Knotilus! Visit <*n*>.<*k*>.html Rolfsen Splice Base's page at the original Knot Atlas! |
[edit] Knot presentations
[edit Notes on presentations of Rolfsen Splice Base]
Computer Talk
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["Rolfsen Splice Base"];
|
In[4]:=
| PD[K]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| Data:Rolfsen Splice Base/PD Presentation |
In[5]:=
| GaussCode[K]
|
Out[5]=
| Data:Rolfsen Splice Base/Gauss Code |
In[6]:=
| DTCode[K]
|
Out[6]=
| Data:Rolfsen Splice Base/DT Code |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
|
In[8]:=
| ConwayNotation[K]
|
Out[8]=
| Data:Rolfsen Splice Base/Conway Notation |
In[9]:=
| br = BR[K]
|
KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
|
Out[9]=
| Data:Rolfsen Splice Base/BraidWord |
In[10]:=
| {First[br], Crossings[br], BraidIndex[K]}
|
KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
|
KnotTheory::loading: Loading precomputed data in IndianaData`.
|
Out[10]=
| { Data:Rolfsen Splice Base/MinimalBraidWidth, Data:Rolfsen Splice Base/MinimalBraidLength, Data:Rolfsen Splice Base/BraidIndex } |
In[11]:=
| Show[BraidPlot[br]]
|
| Data:Rolfsen Splice Base/BraidPlot |
Out[11]=
| -Graphics- |
In[12]:=
| Show[DrawMorseLink[K]]
|
KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
|
KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
|
| Image:Rolfsen Splice Base ML.gif |
Out[12]=
| -Graphics- |
In[13]:=
| ap = ArcPresentation[K]
|
Out[13]=
| ArcPresentationData:Rolfsen Splice Base/Arc Presentation |
In[14]:=
| Draw[ap]
|
| Image:Rolfsen Splice Base AP.gif |
Out[14]=
| -Graphics- |
[edit] Three dimensional invariants
|
[edit Notes for Rolfsen Splice Base's three dimensional invariants] |
[edit] Four dimensional invariants
|
[edit Notes for Rolfsen Splice Base's four dimensional invariants] |
[edit] Polynomial invariants
Further 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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
| K = Knot["Rolfsen Splice Base"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| Data:Rolfsen Splice Base/Alexander Polynomial |
In[5]:=
| Conway[K][z]
|
Out[5]=
| Data:Rolfsen Splice Base/Conway Polynomial |
In[6]:=
| Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
| Data:Rolfsen Splice Base/2nd AlexanderIdeal |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
|
Out[7]=
| { Data:Rolfsen Splice Base/Determinant, Data:Rolfsen Splice Base/Signature } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| Data:Rolfsen Splice Base/Jones Polynomial |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| Data:Rolfsen Splice Base/HOMFLYPT Polynomial |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| Data:Rolfsen Splice Base/Kauffman Polynomial |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {<* alex = Alexander[K][t];
others = DeleteCases[Select[AllKnots[], (alex === Alexander[#][t])&], K];
If[others === {}, "", StringJoin[("[["<>NameString[#]<>"]], ")& /@ others]]
*>}
Same Jones Polynomial (up to mirroring, 
):
{<* J = Jones[K][q];
others = DeleteCases[Select[AllKnots[], (J === Jones[#][q]}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["Rolfsen Splice Base"];
|
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]=
| { Data:Rolfsen Splice Base/Alexander Polynomial, Data:Rolfsen Splice Base/Jones Polynomial } |
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]=
| {<* alex = Alexander[K][t];
others = DeleteCases[Select[AllKnots[], (alex === Alexander[#][t])&], K];
If[others === {}, "", StringJoin[("[["<>NameString[#]<>"]], ")& /@ others]]
*>}
|
In[6]:=
| DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
| {<* J = Jones[K][q];
others = DeleteCases[Select[AllKnots[], (J === Jones[#][q]} |
[edit] Khovanov Homology
| The coefficients of the monomials trqj are shown, along with their alternating sums χ (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 = Data:Rolfsen Splice Base/Signature is the signature of Rolfsen Splice Base. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. | Data:Rolfsen Splice Base/KhovanovTable |
| Integral Khovanov Homology
(db, data source) | Data:Rolfsen Splice Base/Integral Khovanov Homology |
[edit] The Coloured Jones Polynomials
| n | Jn |
| 2 | <*ColouredJones[K, 2][q]*> |
| 3 | <*ColouredJones[K, 3][q]*> |
| 4 | <*ColouredJones[K, 4][q]*> |
| 5 | <*ColouredJones[K, 5][q]*> |
| 6 | <*ColouredJones[K, 6][q]*> |
| 7 | <*ColouredJones[K, 7][q]*> |
[edit] Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.
[edit] Modifying This Page
| Read me first: Modifying Knot Pages
See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top. |
|
