Date of Final Oral Examination (Defense)
Type of Culminating Activity
Master of Science in Mathematics
Jens Harlander, Ph.D.
Zachariah Teitler, Ph.D.
Uwe Kaiser, Ph.D.
Given a polynomial f(x,y) monic in y of degree d, we study the complement ℂ2-C, where C is the curve defined by the equation f(x,y)=0. The Zariski-Van Kampen theorem gives a presentation of the fundamental group of the complement ℂ2-C. Let NT be be the set of complex numbers x for which f(x,y) has multiple roots (as a polynomial in y). Let : ℂ − NT → ℂd − Δ be the map that sends x to the d-tuple of distinct roots (Δ is the diagonal in ℂd). It induces a map ∇: Fr → Bd on the fundamental group level, where Fr is the free group on r letters and Bd is the braid group on d strands. In order to write down the Zariski-Van Kampen presentation one needs an explicit understanding of ∇. This is hard to come by in general. It turns out that under special circumstances ∇ can be computed directly from combinatorial and visual (real) information on the curve C. The method in these special situations is similar to the computation of the presentation of the fundamental group of a knot complement in ℝ3.
Scofield, Mitchell, "On the Fundamental Group of Plane Curve Complements" (2019). Boise State University Theses and Dissertations. 1538.