Knot Invariants from Symbolic Dynamical Systems
نویسندگان
چکیده
If G is the group of an oriented knot k, then the set Hom(K,Σ) of representations of the commutator subgroup K = [G,G] into any finite group Σ has the structure of a shift of finite type ΦΣ, a special type of dynamical system completely described by a finite directed graph. Invariants of ΦΣ, such as its topological entropy or the number of its periodic points of a given period, determine invariants of the knot. When Σ is abelian, ΦΣ gives information about the infinite cyclic cover and the various branched cyclic covers of k. Similar techniques are applied to oriented links.
منابع مشابه
From Time Series to Symbolic Dynamics: An Algebraic Topological Approach
A new approach to constructing a dynamical systems model from experimental time series data is presented. Using the the ideas of delay reconstruction a multivalued dynamical system is constructed. The multivalued approach is taken to allow for bounded experimental error. This system is then analyzed and algebraic invariants based on the Conley index are computed. These invariants have implicati...
متن کاملKnots, Links and Representation Shifts
Introduction. The group π1(S − k) of a knot k contains an extraordinary amount of information. From combined results of W. Whitten [Wh] and M. Culler, C. McA. Gordon, J. Luecke and P.B. Shalen [CuGoLuSh] it is known that there are at most two distinct unoriented prime knots with isomorphic groups. Unfortunately, knot groups are generally difficult to use. Knot groups are usually described by pr...
متن کاملTopological Invariants of Dynamical Systems and Spaces of Holomorphic Maps: I
Departing from the symbolic dynamics, we study natural group action on spaces of holomorphic maps and complex subvarieties. Mathematics Subject Classifications (1991): 32Hxx, 58C10.
متن کاملDynamical generalizations of the Lagrange spectrum
We compute two invariants of topological conjugacy, the upper and lower limits of the inverse of Boshernitzan's ne n , where e n is the smallest measure of a cylinder of length n, for three families of symbolic systems, the natural codings of rotations and three-interval exchanges and the Arnoux-Rauzy systems. The sets of values of these invariants for a given family of systems generalize the L...
متن کاملSymbolic computation with finite biquandles
A method of computing a basis for the second Yang-Baxter cohomology of a finite biquandle with coefficients in Q and Zp from a matrix presentation of the finite biquandle is described. We also describe a method for computing the Yang-Baxter cocycle invariants of an oriented knot or link represented as a signed Gauss code. We provide a URL for our Maple implementations of these algorithms.
متن کامل