Template:Rolfsen Knot Page
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 Rolfsen Knot Page 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]
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
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
|
K = Knot["Rolfsen Knot Page"];
|
In[4]:=
|
Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
Data:Rolfsen Knot Page/Alexander Polynomial |
In[5]:=
|
Conway[K][z]
|
Out[5]=
|
Data:Rolfsen Knot Page/Conway Polynomial |
In[6]:=
|
Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
|
Data:Rolfsen Knot Page/2nd AlexanderIdeal |
In[7]:=
|
{KnotDet[K], KnotSignature[K]}
|
Out[7]=
|
{ Data:Rolfsen Knot Page/Determinant, Data:Rolfsen Knot Page/Signature } |
In[8]:=
|
Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
|
Data:Rolfsen Knot Page/Jones Polynomial |
In[9]:=
|
HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
|
Data:Rolfsen Knot Page/HOMFLYPT Polynomial |
In[10]:=
|
Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
Data:Rolfsen Knot Page/Kauffman Polynomial |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {{{{same_alexander}}}}
Same Jones Polynomial (up to mirroring, ): {{{{same_jones}}}}
KnotTheory`
. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
|
AppendTo[$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}}}} |
Vassiliev invariants
V2 and V3: | (Data:Rolfsen Knot Page/V 2, Data:Rolfsen Knot Page/V 3) |
V2,1 through V6,9: |
V2,1 through V6,9 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 are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 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
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. |
|