2025/12/14 by Denne, Elizabeth
#49Q10 (Secondary) #57K10 (Primary) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2512.12830
Given a thin strip of paper, tie a knot, connect the ends, and flatten into the plane. This is a physical model of a folded ribbon knot in the plane, first introduced by Louis Kauffman. We study the folded ribbonlength of these folded ribbon knots, which is defined as the knot's length-to-width ratio. The \em ribbonlength problem asks to find the infimal folded ribbonlength of a knot or link type. We prove that any P(p,q,r) pretzel link can be constructed so that its infimal folded ribbonlength is ≤ (55)/(√(3)) ≤ 31.755. We prove that any n-strand pretzel link P(p1,p2, …, pn) can be constructed so that its infimal folded ribbonlength is ≤ (18n+1)/(√(3)). This means that there is an infinite link family with a uniform bound on infimal folded ribbonlength. That is, we have shown α=0 in the equation c⋅ Cr(L)α≤ Rib([L]), where L is any link and c is a constant. This paper also contains a table showing the best known upper bounds on the infimal folded ribbonlength for all knots with ≤ 9 crossings.