Identifying Knots within a List

KnotTheory`/Navigation  The Mathematica Package KnotTheory` Acknowledgement Setup Naming and Enumeration Presentations Planar Diagrams Some further details Gauss Codes DT (Dowker-Thistlethwaite) Codes Braid Representatives MorseLink Presentations Arc Presentations Conway Notation Graphical Input Graphical Output Drawing Planar Diagrams How does it work? Drawing MorseLink Presentations Drawing Braids Structure and Operations Invariants Invariants from Braid Theory Three Dimensional Invariants The Alexander-Conway Polynomial The Multivariable Alexander Polynomial The Determinant and the Signature The Jones Polynomial How is the Jones polynomial computed? The Coloured Jones Polynomials The A2 Invariant Quantum knot invariants The HOMFLY-PT Polynomial The Kauffman Polynomial Finite Type (Vassiliev) Invariants Khovanov Homology Heegaard Floer Knot Homology R-Matrix Invariants Extras Included with KnotTheory` Drawing with TubePlot Standalone_TubePlot Using the LinKnot package WikiLink QuantumGroups` Lightly Documented Features A Sample KnotTheory` Session Further Usage Example Prime Links with a Non-Prime Component "Rubberband" Brunnian Links Identifying Knots within a List Cabling Burau's Theorem Further Knot Theory Software KnotPlot Knotscape Xknots Extending/Modifying KnotTheory` Bugs How to Edit this Manual...

IdentifyWithin[L,H], whose code is available here, returns those elements from the list of knots $H$, whose invariant matches that of the knot $L$. It can also recognize mirrors and connected sums of the knots in the list. Its options include turning off (on) the search for connected sums with ConnectedSum->False (True) and choosing the invariants to be used in identification by selecting, for example, Invariants->{Jones[#][q]&, HOMFLYPT[#][a,z]&}. IdentifyWithin can be used together with SubLink to determine the components of a link. For the second component of link L11n150, for instance, we get:

(For In[1] see Setup)

In[2]:=

Import["http://katlas.org/wiki/IdentifyWithin.m&action=raw"];

In[4]:=	IdentifyWithin[SubLink[Link["L11n150"], 2], AllKnots[]]
Out[4]=	{Knot[5, 2]}

L11n150

5_2

Unfortunately, the program does not provide absolute identification when all the used invariants cannot distinguish between two or more different knots. In that case, a list of possible candidates for $L$ is returned.