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]<>" "<>ThisKnot*>'s page] at Knotilus! Visit <*m*>.<*n*>.html <*ThisKnot*>'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 $s=$<*s=KnotSignature[K]*> 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], s+{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[]*>