2024/03/13 by Takeyoshi Kogiso, Kogiso, Takeyoshi, Kengo Miyamoto +7
Computer Science · Mathematics · #05A30 #11A55 #16G20 #57K14 #Advanced Mathematical Identities #Benford’s Law and Fraud Detection #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2403.08446
openalex publication_date 2024/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recently, Morier-Genoud and Ovsienko introduced a q-deformation of rational numbers. More precisely, for an irreducible fraction \fracrs>0, they constructed coprime polynomials R_\fracrs(q),~ S_\fracrs(q) ∈ \mathbb Z[q] with R_\fracrs(1)=r,~S_\fracrs(1)=s. Their theory has a rich background and many applications. By definition, if r ≡ r' \pmods, then S_\fracrs(q)=S_\fracr's(q). We show that rr'≡ -1 \pmods implies S_\fracrs(q)=S_\fracr's(q), and it is conjectured that the converse holds if s is prime (and r \not ≡ r' \pmods). We also show that s is a multiple of 3 (resp. 4) if and only if S_\fracrs(ζ)=0 for ζ=(-1+√(-3))/2 (resp. ζ=i). We give applications to the representation theory of quivers of type A and the Jones polynomials of rational links.