Template:Rolfsen Knot Page
From Knot Atlas
This is the Rolfsen Knot Page master template. For testing purposes, some of the data in it is filled in with data for 8_17.
|
|
|
|
 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit Rolfsen Knot Page's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit Data:Rolfsen Knot Page/KnotilusURL at Knotilus! Visit Rolfsen Knot Page's page at the original Knot Atlas! |
Knot presentations
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||
Length is 8, width is 3, Braid index is 3 | 
|  [{7, 10}, {9, 2}, {10, 4}, {3, 5}, {4, 8}, {6, 9}, {5, 1}, {2, 7}, {1, 6}, {8, 3}] |
[edit Notes on presentations of Rolfsen Knot Page]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["Rolfsen Knot Page"];
|
In[4]:=
| PD[K]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| Data:Rolfsen Knot Page/PD Presentation |
In[5]:=
| GaussCode[K]
|
Out[5]=
| Data:Rolfsen Knot Page/Gauss Code |
In[6]:=
| DTCode[K]
|
Out[6]=
| Data:Rolfsen Knot Page/DT Code |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
|
In[8]:=
| ConwayNotation[K]
|
Out[8]=
| Data:Rolfsen Knot Page/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 Knot Page/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]=
| { 3, 8, 3 } |
In[11]:=
| Show[BraidPlot[br]]
|
|
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."
|
|
|
Out[12]=
| -Graphics- |
In[13]:=
| ap = ArcPresentation[K]
|
Out[13]=
| ArcPresentation[{7, 10}, {9, 2}, {10, 4}, {3, 5}, {4, 8}, {6, 9}, {5, 1}, {2, 7}, {1, 6}, {8, 3}] |
In[14]:=
| Draw[ap]
|
|
|
Out[14]=
| -Graphics- |
Three dimensional invariants
|
[edit Notes for Rolfsen Knot Page's three dimensional invariants] |
Four dimensional invariants
|
[edit Notes for Rolfsen Knot Page's four dimensional invariants] |
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
| K = Knot["Rolfsen Knot Page"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| Data:Rolfsen Knot Page/Alexander Polynomial |
In[5]:=
| Conway[K][z]
|
Out[5]=
| Data:Rolfsen Knot Page/Conway Polynomial |
In[6]:=
| Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
| Data:Rolfsen Knot Page/2nd AlexanderIdeal |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
|
Out[7]=
| { Data:Rolfsen Knot Page/Determinant, Data:Rolfsen Knot Page/Signature } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| Data:Rolfsen Knot Page/Jones Polynomial |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| Data:Rolfsen Knot Page/HOMFLYPT Polynomial |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| Data:Rolfsen Knot Page/Kauffman Polynomial |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {{{{same_alexander}}}}
Same Jones Polynomial (up to mirroring, 
):
{{{{same_jones}}}}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["Rolfsen Knot Page"];
|
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 Knot Page/Alexander Polynomial, Data:Rolfsen Knot Page/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]=
| {{{{same_alexander}}}} |
In[6]:=
| DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
| {{{{same_jones}}}} |
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 Knot Page/Signature is the signature of Rolfsen Knot Page. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. | Data:Rolfsen Knot Page/KhovanovTable |
| Integral Khovanov Homology
(db, data source) | Data:Rolfsen Knot Page/Integral Khovanov Homology |
The Coloured Jones Polynomials
| n | Jn |
| 2 | {{{coloured_jones_2}}} |
| 3 | {{{coloured_jones_3}}} |
| 4 | {{{coloured_jones_4}}} |
| 5 | {{{coloured_jones_5}}} |
| 6 | {{{coloured_jones_6}}} |
| 7 | {{{coloured_jones_7}}} |
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.
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. |
|
