Rolfsen Splice Base: Difference between revisions

Revision as of 22:39, 29 August 2005

Visit <*n*>&id=<*k*> Rolfsen Splice Base's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit [<*KnotilusURL[K]*> Rolfsen Splice Base's page] at Knotilus!

Visit <*n*>.<*k*>.html Rolfsen Splice Base's page at the original Knot Atlas!

Rolfsen Splice Base Further Notes and Views

Knot presentations

Planar diagram presentation	Data:Rolfsen Splice Base/PD Presentation
Gauss code	Data:Rolfsen Splice Base/Gauss Code
Dowker-Thistlethwaite code	Data:Rolfsen Splice Base/DT Code
Conway Notation	Data:Rolfsen Splice Base/Conway Notation

Minimum Braid Representative: {{{braid_table}}}

Length is {{{braid_crossings}}}, width is {{{braid_width}}}.

Braid index is {{{braid_index}}}.

A Morse Link Presentation:

File:Rolfsen Splice Base ML.gif

Three dimensional invariants

Symmetry type	Data:Rolfsen Splice Base/Symmetry Type
Unknotting number	Data:Rolfsen Splice Base/Unknotting Number
3-genus	Data:Rolfsen Splice Base/3-Genus
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:Rolfsen Splice Base/ThurstonBennequinNumber
Hyperbolic Volume	Data:Rolfsen Splice Base/HyperbolicVolume
A-Polynomial	See Data:Rolfsen Splice Base/A-polynomial

[edit Notes for Rolfsen Splice Base's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	Missing
Rasmussen s-Invariant	Missing

[edit Notes for Rolfsen Splice Base's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Data:Rolfsen Splice Base/Alexander Polynomial
Conway polynomial	Data:Rolfsen Splice Base/Conway Polynomial
2nd Alexander ideal (db, data sources)	Data:Rolfsen Splice Base/2nd AlexanderIdeal
Determinant and Signature	{ Data:Rolfsen Splice Base/Determinant, Data:Rolfsen Splice Base/Signature }
Jones polynomial	Data:Rolfsen Splice Base/Jones Polynomial
HOMFLY-PT polynomial (db, data sources)	Data:Rolfsen Splice Base/HOMFLYPT Polynomial
Kauffman polynomial (db, data sources)	Data:Rolfsen Splice Base/Kauffman Polynomial
The A2 invariant	Data:Rolfsen Splice Base/QuantumInvariant/A2/1,0
The G2 invariant	Data:Rolfsen Splice Base/QuantumInvariant/G2/1,0

Further Quantum Invariants

Rolfsen Splice Base 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

"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, $q\leftrightarrow q^{-1}$ ): {<* J = Jones[Knot[n,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[Knot[n,k]][q];

                       others = DeleteCases[Select[AllKnots[], (J === Jones[#][q]}

Vassiliev invariants

V₂ and V₃:

(Data:Rolfsen Splice Base/V 2, Data:Rolfsen Splice Base/V 3)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

Data:Rolfsen Splice Base/V 2,1

Data:Rolfsen Splice Base/V 3,1

Data:Rolfsen Splice Base/V 4,1

Data:Rolfsen Splice Base/V 4,2

Data:Rolfsen Splice Base/V 4,3

Data:Rolfsen Splice Base/V 5,1

Data:Rolfsen Splice Base/V 5,2

Data:Rolfsen Splice Base/V 5,3

Data:Rolfsen Splice Base/V 5,4

Data:Rolfsen Splice Base/V 6,1

Data:Rolfsen Splice Base/V 6,2

Data:Rolfsen Splice Base/V 6,3

Data:Rolfsen Splice Base/V 6,4

Data:Rolfsen Splice Base/V 6,5

Data:Rolfsen Splice Base/V 6,6

Data:Rolfsen Splice Base/V 6,7

Data:Rolfsen Splice Base/V 6,8

Data:Rolfsen Splice Base/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=

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

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the

n+1

-dimensional representation of

sl(2)

)

$n$	$J_{n}$
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]>

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

<*InOut["PD[``]", K]*>

<*InOut["GaussCode[``]", K]*>

<*InOut["DTCode[``]", K]*>

<*InOut["br = BR[``]", K]*>

<*InOut["{First[br], Crossings[br]}"]*>

<*InOut["BraidIndex[``]", K]*>

<*GraphicsBox["`1`_`2`_ML.gif", "Show[DrawMorseLink[Knot[`1`, `2`]]]", n, k]*>

<*InOut[

 "(#[``]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}",  K

]*>

<*InOut["alex = Alexander[``][t]", K]*>

<*InOut["Conway[``][z]", K]*>

<*InOut["Select[AllKnots[], (alex === Alexander[#][t])&]"]*>

<*InOut["{KnotDet[`1`], KnotSignature[`1`]}", K]*>

<*InOut["Jones[``][q]", K]*>

<*InOut[

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

]*>

<*InOut["A2Invariant[``][q]", K]*>

<*InOut["HOMFLYPT[``][a, z]", K]*>

<*InOut["Kauffman[``][a, z]", K]*>

<*InOut["{Vassiliev[2][`1`], Vassiliev[3][`1`]}", K]*>

<*InOut["Kh[``][q, t]", K]*>

<* If[ColouredJones[K, 2] === NotAvailable, "",

 InOut["ColouredJones[``, 2][q]", K]

] *>

In[1]:=	<< KnotTheory`
<InOut[1]; KnotTheoryWelcomeMessage[]>

See/edit the Rolfsen_Splice_Template.

Back to the top.

[[Image:Data:Rolfsen Splice Base/Previous Knot.gif|80px|link=Data:Rolfsen Splice Base/Previous Knot]]

[[Data:Rolfsen Splice Base/Previous Knot]]

[[Image:Data:Rolfsen Splice Base/Next Knot.gif|80px|link=Data:Rolfsen Splice Base/Next Knot]]

[[Data:Rolfsen Splice Base/Next Knot]]

<* (* *) *>

@@ Line 29: / Line 29: @@
 {{Similar Knots|
-     same_alexander=<*  alex = Alexander[K][t];
+     same_alexander= <*  alex = Alexander[K][t];
-                        others = DeleteCases[Select[AllKnots[], (alex === Alexander[#][t])&], K];
+                         others = DeleteCases[Select[AllKnots[], (alex === Alexander[#][t])&], K];
+                         If[others === {}, "", StringJoin[("[["<>NameString[#]<>"]], ")& /@ others]]
+                     *> |
+     same_jones=     <* J = Jones[Knot[n,k]][q];
+                        others = DeleteCases[Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&], K];
                         If[others === {}, "", StringJoin[("[["<>NameString[#]<>"]], ")& /@ others]]
-                    *> |
+                     *>}}
-     same_jones=<* J = Jones[Knot[n,k]][q];
-                   others = DeleteCases[Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&], K];
-                   If[others === {}, "", StringJoin[("[["<>NameString[#]<>"]], ")& /@ others]]
-                *>}}
 {{Vassiliev Invariants}}
 {{Khovanov Homology|table=<*TabularKh[Kh[K][q, t], KnotSignature[K]+{1,-1}]*>}}
+{{Display Coloured Jones|
-{{Display Coloured Jones|J2=<*ColouredJones[K, 2][q]*>|J3=<*ColouredJones[K, 3][q]*>|J4=<*ColouredJones[K, 4][q]*>|J5=<*ColouredJones[K, 5][q]*>|J6=<*ColouredJones[K, 6][q]*>|J7=<*ColouredJones[K, 7][q]*>}}
+   J2 = <*ColouredJones[K, 2][q]*> |
+   J3 = <*ColouredJones[K, 3][q]*> |
+   J4 = <*ColouredJones[K, 4][q]*> |
+   J5 = <*ColouredJones[K, 5][q]*> |
+   J6 = <*ColouredJones[K, 6][q]*> |
+   J7 = <*ColouredJones[K, 7][q]*>   }}
 {{Computer Talk Header}}

$n$	$J_{n}$
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]>

Rolfsen Splice Base: Difference between revisions

Revision as of 22:39, 29 August 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

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