In order to work with braids using Conway symbols of KLs as an input, first you need to convert the input by using the LinKnot function fCreatePData, and then work with P-data obtained as before. Then you can start making experiments with KLs in order to try to find some connections between Conway symbols and braid words. For example, every KL of the KL family p (p = 2,3,...) has a braid word Ap, every KL of the family p 1 2 (p = 2,3,...) has a braid word ApbAb, to every KL of the family p 1 1 q corresponds a braid word ApbAbq, etc. Indeed, general formulas of this kind can be obtained for all kinds of KL families, not just for rational KLs.
From braid words you can calculate different KL invariants. The LinKnot function fSeifert (webMathematica fSeifert) calculates the Seifert matrix of a KL given by its Conway symbol, Dowker code, or P-data. The function fSignat (webMathematica fSignat) calculates the signature of a KL given by its Conway symbol, Dowker code, or P-data. Their basic functions ssmW and SeifertMatrix were written by S. Orevkov.