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

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 $b(L) \leq \frac{1}{2} c(L) + 1$and$\alpha(L) \leq c(L) + 2$. We also find a quadratic upper bound of the minimal crossing number of delta diagrams of $L$.