On $q$-deformed Farey sum and a homological interpretation of $q$-deformed real quadratic irrational numbers

TL;DR

This paper introduces a new formula for $q$-deformed Farey sums based on negative continued fractions, linking them to knot invariants and homological algebra, and explores their relation to real quadratic irrationals.

Contribution

It provides a combinatorial formula for $q$-deformed Farey sums and connects these to homological interpretations and knot invariants, extending previous work on $q$-deformed rational numbers.

Findings

Derived a formula for $q$-deformed Farey sum using negative continued fractions.

Established a combinatorial proof relating $q$-deformed rational numbers to Jones polynomials.

Proposed a homological interpretation of the $q$-deformed Farey sum and its connection to real quadratic irrationals.

Abstract

The left and right $q$-deformed rational numbers were introduced by Bapat, Becker and Licata via regular continued fractions, and they gave a homological interpretation for left and right $q$-deformed rational numbers. In the present paper, we focus on negative continued fractions and defined left $q$-deformed negative continued fractions. We give a formula for computing the $q$-deformed Farey sum of the left $q$-deformed rational numbers based on it. We use this formula to give a combinatorial proof of the relationship between the left $q$-deformed rational number and the Jones polynomial of the corresponding rational knot which was proved by Bapat, Becker and Licata using a homological technique. Finally, we combine their work and the $q$-deformed Farey sum, and give a homological interpretation of the $q$-deformed Farey sum. We also give an approach to finding a relationship between real quadratic irrational numbers and homological algebra.