Bisected vertex leveling of plane graphs: Braid index, arc index and delta diagrams

Abstract

In this paper, we introduce a bisected vertex leveling of a plane graph. Using this planar embedding, we present elementary proofs of the well-known upper bounds in terms of the minimal crossing number on braid index b(L) and arc index alpha(L) for any knot or non-split link L, which are b(L) <= 1/2c(L) + 1 and alpha(L) <= c(L) + 2. We also find a quadratic upper bound of the minimal crossing number of delta diagrams of L.