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Topologies on sets of polynomial knots and the homotopy types of the respective spaces
by
Hitesh Raundal
Harish-Chandra Research Institute, Prayagraj (Allahabad), India

A polynomial knot is a smooth embedding R\hookrightarrowRⁿ such that the component functions are real polynomials. Let Pⁿ be the set of all polynomial knots in Rⁿ, and let P=∪_{n ∈ Z⁺}Pⁿ. By identifying a polynomial knot φ:R\hookrightarrowRⁿ\hookrightarrowR^∞ with a Λ-tuple (φ_ij)_{i, j} (where Λ = Z⁺×N and φ(t)=∑_j(φ_ij)_it^j for t ∈ R), we can think of the sets Pⁿ and P as subsets of R^Λ, and thus they can be given the subspace topologies that inherit from the box and the product topologies of R^Λ. We also have the topologies on Pⁿ and P induced by the metrics d_r (for r ≥ 1) and d_∞ given by d_r(φ, ψ)=(∑_{i, j}|φ_ij-ψ_ij|^r)^1/r and d_∞(φ, ψ)=sup_{i, j} | φ_ij-ψ_ij| for φ, ψ ∈ P. We show that Pⁿ has the same homotopy type as S^n-1 and P is contractible, where the spaces have any of the topologies described above.

Date received: November 10, 2018

Copyright © 2018 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbpq-09.