2018/09/25 by Hitesh Raundal, Raundal, Hitesh
Mathematics · #2010: Primary 57M25 #55P10 #55P15 #57R40 #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #Secondary 54A10
paper · pdf · doi:10.48550/arxiv.1810.00666
openalex publication_date 2018/09/25 · openalex created_date 2022/08/03 · openalex updated_date 2026/07/28
A polynomial knot in \ℝn is a smooth embedding of \ℝ in\n\ℝn such that the component functions are real polynomials. In the\nearlier paper with Mishra, we have studied the space \P of\npolynomial knots in \ℝ3 with the inductive limit topology coming\nfrom the spaces \Od for d\≥3, where \Od is the\nspace of polynomial knots in \ℝ3 with degree d and having some\nconditions on the degrees of the component polynomials. In the same paper, we\nhave proved that the space of polynomial knots in \ℝ3 has the same\nhomotopy type as S2. The homotopy type of the space is the mere consequence\nof the topology chosen. If we have another topology on \P, the\nhomotopy type may change. With this in mind, we consider in general the set\n\Ln of polynomial knots in \ℝn with various topologies\non it and study the homotopy type of the respective spaces. Let \L\nbe the union of the sets \Ln for n\≥1. We also explore the\nhomotopy type of the space \L with some natural topologies on it.\n