Abstract
In [B], Birman proposed (but considered as very difficult) the problem to classify 3-braid links with given polynomials. In [St2] we dealt with the skein polynomial. Now we can extend our results to the Alexander polynomial (with the convention in the beginning of Section 3.3). The following discussion gives a fairly exact description how to find the 3-braid links, if such exist, for any possible admissible (as specified in Section 2.2) polynomial.
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- 1.
Note that here the brackets for polynomials and for braids have a completely different meaning.
- 2.
Note that these are exactly the 3-braids which are reducible in the terminology of dynamic properties; this reducibility has nothing to do, though, with the reducibility in Markov’s theorem.
- 3.
Note that V is here what is written as J, and not V, in [K].
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Stoimenow, A. (2017). Polynomial Invariants. In: Properties of Closed 3-Braids and Braid Representations of Links . SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-68149-8_4
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