The braid indices of pretzel links: A comprehensive study, part II

This paper is the second installment of our in-depth study on the braid index problem of pretzel links. We classify pretzel links into three categories based on the Seifert circle decompositions in their standard link diagrams. In our initial paper, we successfully identified the braid indices for all Type 1 and Type 2 pretzel links. In this paper, we share our findings on Type 3 pretzel links. We are able to determine the exact braid indices for most Type 3 pretzel links. For the small subset of Type 3 pretzel links where exact braid indices remain undetermined, we establish their braid indices within a range of two consecutive integers. With some numerical evidence, we conjecture that in such an exceptional case, the braid index of the Type 3 pretzel link is given by the larger of the two consecutive integers given by our formulas.