Torus Knot Splice Base

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

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

File:Torus Knot Splice Base.jpg

Visit [<*KnotilusURL[K]*> Torus Knot Splice Base's page] at Knotilus! Visit <*m*>.<*n*>.html Torus Knot Splice Base's page at the original Knot Atlas! Torus Knot Splice Base Quick Notes

Torus Knot Splice Base Further Notes and Views

Knot presentations

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

Knot presentations

Planar diagram presentation	<*PD[K]*>
Gauss code	<*List @@ GaussCode[K]*>
Dowker-Thistlethwaite code	<*StringReplace[StringTake[ToString[DTCode[K]], {8, -2}], ","->""]*>

Polynomial invariants

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

Further Quantum Invariants

Torus Knot 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["Torus Knot Splice Base"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

Data:Torus Knot Splice Base/Alexander Polynomial

In[5]:=

Conway[K][z]

Out[5]=

Data:Torus Knot 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:Torus Knot Splice Base/2nd AlexanderIdeal

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ Data:Torus Knot Splice Base/Determinant, Data:Torus Knot Splice Base/Signature }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

Data:Torus Knot Splice Base/Jones Polynomial

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

Data:Torus Knot Splice Base/HOMFLYPT Polynomial

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

Data:Torus Knot Splice Base/Kauffman Polynomial

Vassiliev invariants

V2 and V3:

(Data:Torus Knot Splice Base/V 2, Data:Torus Knot 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:Torus Knot Splice Base/V 2,1 Data:Torus Knot Splice Base/V 3,1 Data:Torus Knot Splice Base/V 4,1 Data:Torus Knot Splice Base/V 4,2 Data:Torus Knot Splice Base/V 4,3 Data:Torus Knot Splice Base/V 5,1 Data:Torus Knot Splice Base/V 5,2 Data:Torus Knot Splice Base/V 5,3 Data:Torus Knot Splice Base/V 5,4 Data:Torus Knot Splice Base/V 6,1 Data:Torus Knot Splice Base/V 6,2 Data:Torus Knot Splice Base/V 6,3 Data:Torus Knot Splice Base/V 6,4 Data:Torus Knot Splice Base/V 6,5 Data:Torus Knot Splice Base/V 6,6 Data:Torus Knot Splice Base/V 6,7 Data:Torus Knot Splice Base/V 6,8 Data:Torus Knot 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:Torus Knot Splice Base/Signature is the signature of Torus Knot Splice Base. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

<*TabularKh[Kh[K][q, t], KnotSignature[K]+{1,-1}]*>

Computer Talk

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

<*InOut["Crossings[``]", K]*> <*InOut["PD[``]", K]*> <*InOut["GaussCode[``]", K]*> <*InOut["BR[``]", K]*> <*InOut["alex = Alexander[``][t]", K]*> <*InOut["Conway[``][z]", K]*> <*InOut["Select[AllKnots[], (alex === Alexander[#][t])&]"]*> <*InOut["{KnotDet[`1`], KnotSignature[`1`]}", K]*> <*InOut["J=Jones[``][q]", K]*> <*InOut[

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

]*> <* If[Crossings[K]<=18, Include["ColouredJonesM.mhtml"] ,""] *> <*InOut["A2Invariant[``][q]", K]*> <*InOut["Kauffman[``][a, z]", K]*> <*InOut["{Vassiliev[2][`1`], Vassiliev[3][`1`]}", K ]*> <*InOut["Kh[``][q, t]", K]*>

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