#### Publication Date

5-2019

#### Date of Final Oral Examination (Defense)

3-1-2019

#### Type of Culminating Activity

Thesis

#### Degree Title

Master of Science in Mathematics

#### Department

Mathematics

#### Major Advisor

Jens Harlander, Ph.D.

#### Advisor

Zachariah Teitler, Ph.D.

#### Advisor

Uwe Kaiser, Ph.D.

#### Abstract

Given a polynomial *f*(*x*,*y*) monic in *y* of degree *d*, we study the complement ℂ^{2}-*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 ℂ^{2}-*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* → ℂ* ^{d}* − Δ be the map that sends

*x*to the

*d*-tuple of distinct roots (Δ is the diagonal in ℂ

^{d}). It induces a map ∇:

*F*

_{r}→

*B*on the fundamental group level, where

_{d}*F*is the free group on

_{r}*r*letters and

*B*is the braid group on

_{d}*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 ℝ

^{3}.

#### DOI

10.18122/td/1538/boisestate

#### Recommended Citation

Scofield, Mitchell, "On the Fundamental Group of Plane Curve Complements" (2019). *Boise State University Theses and Dissertations*. 1538.

10.18122/td/1538/boisestate