2025/09/22 by Denne, Elizabeth, Patterson, Timi · 2 citations
#FOS: Mathematics #Geometric Topology (math.GT) #Primary 57K10 #Secondary 49Q10
paper · doi:10.48550/arxiv.2509.18370
Take a thin, rectangular strip of paper, add in an odd number of half-twists, then join the ends together. This gives a multi-twist paper Möbius band. We prove that any multi-twist paper Möbius band can be constructed so the aspect ratio of the rectangle is 3√(3)+ε for any ε>0. We could also take the thin, rectangular strip of paper and tie a knot in it, then join the ends and fold flat in the plane. This creates a folded ribbon knot. We apply the techniques used to prove the multi-twist paper Möbius band result to (2,q) torus knots and twist knots. We prove that any (2,q)-torus knot can be constructed so that the folded ribbonlength ≤ 13.86. We prove that any twist knot can be constructed so that the folded ribbonlength is ≤ 17.59. Both of these results give the lower bound for the ribbonlength crossing number problem which relates the infimal folded ribbonlength of a knot type [K] to its crossing number Cr(K). That is, we have shown α=0 in the equation c⋅ Cr(K)α≤ Rib([K]), where c is a constant.