Surfaces in Seifert fibered spaces Jennifer Schultens University of California, Davis June 26, 2019 Jennifer Schultens Surfaces in Seifert fibered spaces We also study submanifolds. E.g., 1-dimensional knots in 3-dimensional space. I will discuss the totality of surfaces in a particular class of 3-manifolds. Rather than giving you a formal definition of manifold, in particular 3-manifold, I will describe some of my favorite examples. General comments Low-dimensional topology concerns the topology of manifolds in 1, 2, 3, and 4 dimensions. Jennifer Schultens Surfaces in Seifert fibered spaces I will discuss the totality of surfaces in a particular class of 3-manifolds. Rather than giving you a formal definition of manifold, in particular 3-manifold, I will describe some of my favorite examples. General comments Low-dimensional topology concerns the topology of manifolds in 1, 2, 3, and 4 dimensions. We also study submanifolds. E.g., 1-dimensional knots in 3-dimensional space. Jennifer Schultens Surfaces in Seifert fibered spaces Rather than giving you a formal definition of manifold, in particular 3-manifold, I will describe some of my favorite examples. General comments Low-dimensional topology concerns the topology of manifolds in 1, 2, 3, and 4 dimensions. We also study submanifolds. E.g., 1-dimensional knots in 3-dimensional space. I will discuss the totality of surfaces in a particular class of 3-manifolds. Jennifer Schultens Surfaces in Seifert fibered spaces General comments Low-dimensional topology concerns the topology of manifolds in 1, 2, 3, and 4 dimensions. We also study submanifolds. E.g., 1-dimensional knots in 3-dimensional space. I will discuss the totality of surfaces in a particular class of 3-manifolds. Rather than giving you a formal definition of manifold, in particular 3-manifold, I will describe some of my favorite examples. Jennifer Schultens Surfaces in Seifert fibered spaces Examples of manifolds: Not knot Definition Let K be a knot. The complement of K is 3 C(K) = S − η(K) where η(K) is a regular neighborhood of K. Jennifer Schultens Surfaces in Seifert fibered spaces Not knot Figure: The complement of the unknot Jennifer Schultens Surfaces in Seifert fibered spaces Not knot Figure: The complement of a knot Jennifer Schultens Surfaces in Seifert fibered spaces Torus knots Figure: T(2, 3), also known as the trefoill Jennifer Schultens Surfaces in Seifert fibered spaces Satellite knots I-’ 4 KC sot’? kftco V i2 (Jattant KcV • Figure: Satellite knots Jennifer Schultens Surfaces in Seifert fibered spaces A proof can be found in M. Kapovich's \Hyperbolic manifolds and discrete groups". Thurston's theorem Theorem 3 (Trichotomy for knots) Let K be a knot in S . Then exactly one of the following holds: K is a torus knot; K is a satellite knot; K is hyperbolic. Jennifer Schultens Surfaces in Seifert fibered spaces Thurston's theorem Theorem 3 (Trichotomy for knots) Let K be a knot in S . Then exactly one of the following holds: K is a torus knot; K is a satellite knot; K is hyperbolic. A proof can be found in M. Kapovich's \Hyperbolic manifolds and discrete groups". Jennifer Schultens Surfaces in Seifert fibered spaces Examples of manifolds: 2-fold branched cover 2-fold branched cover of a knot: The 2-fold branched cover of a knot has two points for every point in the knot complement and one point for every point on the knot. Jennifer Schultens Surfaces in Seifert fibered spaces Examples of manifolds: 2-fold branched cover Figure: A Seifert surface Theorem (Seifert) Every knot admits a Seifert surface. Jennifer Schultens Surfaces in Seifert fibered spaces Examples of manifolds: Seifert fibered spaces Seifert fibered spaces are a family of 3-dimensional spaces (3-manifolds) whose members are classified by a finite set of invariants. The structure of these manifolds allows us to concretely describe essential surfaces embedded in them. We are interested in the collection of all surfaces in a given Seifert fibered space. However, we must restrict our attention to essential surfaces in Seifert fibered spaces. The Kakimizu complex of a knot provides an example of how to encode surfaces. We discus the analogous concept for Seifert fibered spaces. We will always assume that curves in surfaces are simple and that surfaces in 3-manifolds are properly embedded. Jennifer Schultens Surfaces in Seifert fibered spaces Fibered solid tori identify after twist Figure: A fibered solid torus Jennifer Schultens Surfaces in Seifert fibered spaces The fibered solid torus V (ν; µ) is obtained from the solid cylinder D2 × [0; 1] by gluing D2 × f0g to D2 × f1g after rotating D2 2πν about its center point x by µ . 2 Intervals of the form y × [0; 1] for y 2 D match up to form simple closed curves called fibers. All fibers except the one determined by x × I traverse the torus µ times. The curve x × S1 is called the central fiber. If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber. The other fibers are called regular fibers. If µ = 1 or ν = 0, then the central fiber is also called a regular fiber. Two fibered solid tori are considered equivalent if they are homeomorphic via a fiber preserving homeomorphism. Fibered solid tori Let ν 2 Z and µ 2 N be such that g:c:d:(ν; µ) = 1. Jennifer Schultens Surfaces in Seifert fibered spaces 2 Intervals of the form y × [0; 1] for y 2 D match up to form simple closed curves called fibers. All fibers except the one determined by x × I traverse the torus µ times. The curve x × S1 is called the central fiber. If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber. The other fibers are called regular fibers. If µ = 1 or ν = 0, then the central fiber is also called a regular fiber. Two fibered solid tori are considered equivalent if they are homeomorphic via a fiber preserving homeomorphism. Fibered solid tori Let ν 2 Z and µ 2 N be such that g:c:d:(ν; µ) = 1. The fibered solid torus V (ν; µ) is obtained from the solid cylinder D2 × [0; 1] by gluing D2 × f0g to D2 × f1g after rotating D2 2πν about its center point x by µ . Jennifer Schultens Surfaces in Seifert fibered spaces All fibers except the one determined by x × I traverse the torus µ times. The curve x × S1 is called the central fiber. If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber. The other fibers are called regular fibers. If µ = 1 or ν = 0, then the central fiber is also called a regular fiber. Two fibered solid tori are considered equivalent if they are homeomorphic via a fiber preserving homeomorphism. Fibered solid tori Let ν 2 Z and µ 2 N be such that g:c:d:(ν; µ) = 1. The fibered solid torus V (ν; µ) is obtained from the solid cylinder D2 × [0; 1] by gluing D2 × f0g to D2 × f1g after rotating D2 2πν about its center point x by µ . 2 Intervals of the form y × [0; 1] for y 2 D match up to form simple closed curves called fibers. Jennifer Schultens Surfaces in Seifert fibered spaces If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber. The other fibers are called regular fibers. If µ = 1 or ν = 0, then the central fiber is also called a regular fiber. Two fibered solid tori are considered equivalent if they are homeomorphic via a fiber preserving homeomorphism. Fibered solid tori Let ν 2 Z and µ 2 N be such that g:c:d:(ν; µ) = 1. The fibered solid torus V (ν; µ) is obtained from the solid cylinder D2 × [0; 1] by gluing D2 × f0g to D2 × f1g after rotating D2 2πν about its center point x by µ . 2 Intervals of the form y × [0; 1] for y 2 D match up to form simple closed curves called fibers. All fibers except the one determined by x × I traverse the torus µ times. The curve x × S1 is called the central fiber. Jennifer Schultens Surfaces in Seifert fibered spaces The other fibers are called regular fibers. If µ = 1 or ν = 0, then the central fiber is also called a regular fiber. Two fibered solid tori are considered equivalent if they are homeomorphic via a fiber preserving homeomorphism. Fibered solid tori Let ν 2 Z and µ 2 N be such that g:c:d:(ν; µ) = 1. The fibered solid torus V (ν; µ) is obtained from the solid cylinder D2 × [0; 1] by gluing D2 × f0g to D2 × f1g after rotating D2 2πν about its center point x by µ . 2 Intervals of the form y × [0; 1] for y 2 D match up to form simple closed curves called fibers. All fibers except the one determined by x × I traverse the torus µ times. The curve x × S1 is called the central fiber. If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber. Jennifer Schultens Surfaces in Seifert fibered spaces Two fibered solid tori are considered equivalent if they are homeomorphic via a fiber preserving homeomorphism. Fibered solid tori Let ν 2 Z and µ 2 N be such that g:c:d:(ν; µ) = 1. The fibered solid torus V (ν; µ) is obtained from the solid cylinder D2 × [0; 1] by gluing D2 × f0g to D2 × f1g after rotating D2 2πν about its center point x by µ .