Quantum Numbers and q-Deformed Conway–Coxeter Friezes

Abstract

To the memory of John Conway T T he notion of q-deformed real numbers was recently introduced in [15,16]. The first steps of this theory were largely influenced by John Conway. In the fall of 2013, we had a chance to spend a week in his company. The occasion was a conference that we organized in Luminy, at which Conway gave two wonderful talks. We cannot forget the long conversations we had with John there; between desperate attempts to teach us his famous doomsday algorithm, he told us much about his work with Coxeter. Our understanding of Conway and Coxeter's theorem has transformed into a certain combinatorial viewpoint on continued fractions [14] that eventually led us to q-deformations.One goal of this article is to explain the origins of qnumbers and sketch their main properties. We will also introduce q-deformed Conway-Coxeter friezes. We would have loved to have had the chance to talk all of this over with John.John Conway and the authors.

Towards quantized complex numbers: $q$-deformed Gaussian integers and the Picard group

“…We already have the q-deformed operators T q , S q , U q . The remaining generator L and its q-deformation also appeared in the context of q-deformed rational numbers [11,13,9]. The operator of linear-fractional transformations associated with the matrix L is the "negation operator": L(x) = −x.…”

“…The operator of linear-fractional transformations associated with the matrix L is the "negation operator": L(x) = −x. It was observed in [11,13,9] that, besides the invariance under the modular group action, q-deformed rational numbers satisfy one more invariance property:…”

“…It was further extended to arbitrary real numbers in [12]. Several properties of q-numbers were studied in [13,9,10]. The unimodality conjecture formulated in [11] was tackled in [1].…”

“…The notion of q-rationals was defined in [11] in a combinatorial way. It became clear later (see [9,13,10]) that the simplest, and perhaps most conceptual, way to define q-deformed rationals is to assume the modular invariance.…”

On Radius of Convergence of q-Deformed Real Numbers

“…In this paper we discuss a possible q-analogue of this classical result. Namely, we consider the qdeformations (or "q-analogues") of real numbers, which have been recently introduced in [13,14] and studied further in [11,15,9]. They have several nice properties and connections, including theory of Conway-Coxeter friezes and knot invariants.…”