Rolfsen Splice Base

1. REDIRECT Template:Splice Base Notice

[[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]]

File:Rolfsen Splice Base.gif

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 Quick Notes

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: <* BraidPlot[CollapseBraid[br=BR[K]], Mode -> "Wiki", Images -> {"BraidPart0.gif", "BraidPart1.gif", "BraidPart2.gif", "BraidPart3.gif", "BraidPart4.gif"}] *> Length is <*Crossings[br]*>, width is <*First[br]*>. Braid index is <*BraidIndex[K]*>.

A Morse Link Presentation:

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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

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] || (J /. q-> 1/q) === Jones[#][q])&
 ], K];
 If[others === {}, "",
   StringJoin[("[["<>NameString[#]<>"]], ")& /@ others]
 ]

• >...}

Vassiliev invariants

V2 and V3:

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

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 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[``], KnotSignature[``]}", 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][``], Vassiliev[3][``]}", 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.

<* (* *) *>