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In this paper we represent the classical braids in the Yokonuma–Hecke and the adelic Yokonuma–Hecke algebras. More precisely, we define the completion of the framed braid group and we introduce the adelic Yokonuma–Hecke algebras, in analogy to the p-adic framed braids and the p-adic Yokonuma–Hecke algebras introduced in Juyumaya and Lambropoulou (Topol. Appl. 154:1804–1826, 2007; arXiv:0905.3626v1, 2009). We further construct an adelic Markov trace, analogous to the p-adic Markov trace constructed in Juyumaya and Lambropoulou (arXiv:0905.3626v1, 2009), and using the traces in Juyumaya (J. Knot Theory Ramif. 13:25–29, 2004) and the adelic Markov trace we define topological invariants of classical knots and links, upon imposing some condition. Each invariant satisfies a cubic skein relation coming from the Yokonuma–Hecke algebra.

An Adelic Extension of the Jones Polynomial