Broadway Belles

https://digitalcommons.library.umaine.edu/mmb-vp/5076/thumbnail.jp

Link projections and flypes

Let \Pi be a link projection in S^2. John Conway and later Francis Bonahon and Larry Siebenmann undertook to split $\Pi$ into canonical pieces. These pieces received different names: basic or polyhedral diagrams on one hand, rational, algebraic, bretzel, arborescent diagrams on the other hand. This paper proposes a thorough presentation of the theory, known to happy fews. We apply the existence and uniqueness theorem for the canonical decomposition to the classification of Haseman circles and to the localisation of the flypes

On the monodromies of a polynomial map from C2 to C

AbstractLet f:C2→C be a polynomial function. It is well known that there exists a finite set A⊂C such that the restriction of f to C2−f−1(A) is a differentiable fibration onto C−A. Following Broughton in (Proc. Symp. Pure Math. 40 (1983) 167) we call the smallest of such A's the set of atypical values of f and write it Af. Let F be a generic fiber of f. The main goal of this article is to describe the monodromy on H1(F,Z) around an atypical value a∈Af. For that purpose we define and study a monodromic filtration on the homology with coefficients in Z:0⊂M−1⊂M0⊂M1⊂M2=H1(F,Z). The term M−1 is added to allow for the boundary of F. We introduce a compact model L̂a for the smooth part of the reduced curve associated to the affine fiber f−1(a). One important result of this article is theorem (8.12) which shows how H1(L̂a,Z) gives (via the transfer homomorphism) a precise description of the invariant cycles in H1(F,Z)