Jones polynomial of knots formed by repeated tangle replacement operations

In this paper, we prove that the Jones polynomial of a link diagram obtained through repeated tangle replacement operations can be computed by a sequence of suitable variable substitutions in simpler polynomials. For the case that all the tangles involved in the construction of the link diagram have at most k crossings (where k is a constant independent of the total number n of crossings in the link diagram), we show that the computation time needed to calculate the Jones polynomial of the link diagram is bounded above by O ( n k ) . In particular, we show that the Jones polynomial of any Conway algebraic link diagram with n crossings can be computed in O ( n 2 ) time. A consequence of this result is that the Jones polynomial of any Montesinos link and two bridge knot or link of n crossings can be computed in O ( n 2 ) time.