On twist positive braids with the 2-variable link invariant: 3-braids of width 2

Abstract

The possible [Formula: see text]-variable link polynomial [Formula: see text] for an oriented link [Formula: see text], which has width [Formula: see text] in the variable [Formula: see text] is studied, where width of [Formula: see text] is the minimal number of strands allowed by the index bound. It is shown that if the 2-variable link invariant [Formula: see text] for an oriented link [Formula: see text] has width [Formula: see text] in the variable [Formula: see text] then it is the same as the polynomial of a closed [Formula: see text]-braid, [Formula: see text] Also a complete list of [Formula: see text]- braids of width [Formula: see text] which close to knots, are given. Consequently it is shown that [Formula: see text] determines [Formula: see text] for the full [Formula: see text]-braid [Formula: see text] Finally, the 2-variable polynomial for the non-full 3-braid is calculated.