The distribution of braid indices of 2-bridge knots

In this paper, we study the braid indices of [Formula: see text]-bridge knots with a fixed crossing number [Formula: see text]. We show that the average braid index of the set of [Formula: see text]-bridge knots of crossing number [Formula: see text] is asymptotically linear, approaching [Formula: see text]. Additionally, we show that the variance of the braid indices of the set of [Formula: see text]-bridge knots of crossing number [Formula: see text] is also asymptotically linear, approaching [Formula: see text]. Finally, we find a formula for the number [Formula: see text] of [Formula: see text]-bridge knots with crossing number [Formula: see text] and braid index [Formula: see text], and show that for any fixed [Formula: see text], the braid index where [Formula: see text] achieves its maximum is [Formula: see text].