A long-standing problem in knot theory concerns the additivity of crossing numbers of links under the connected sum operation. It is conjectured that if L1 and L2 are links, then Cr(L1#L2)=Cr(L1)+Cr(L2), but so far this has been proved only for certain classes of links. For example, in cases where both L1 and L2 are alternating or adequate links, the conjecture is known to be true. Another situation in which Cr(L1#L2)=Cr(L1)+Cr(L2) is when both L1 and L2 are zero-deficiency links. Zero-deficiency links include some but not all of the links in the prior named classes, as well as some links that are not included in either of those. In addition, further results are known for situations in which only one of the links being connected has deficiency zero. In this paper we expand the known realm of zero-deficiency links to include some cases of links represented by alternating closed braids. The ultimate goal is to show that if Dk is any k-string, reduced, alternating, closed braid, then the braid index of Dk is k. Herein we show the result for a certain subset of these closed braids, those with at most two sequences of crossings between consecutive strings in the braid. This result is proved using a property of the HOMFLY polynomial, which provides a lower bound for the braid index of a link. In the process, a simplified formulation for computing the HOMFLY polynomial is implemented. It seems likely that this result can be extended to prove the result for more complex alternating closed braids.
