Decomposition into 3-Balls
In this chapter, we start with a connected link diagram and explain how to construct state graphs and state surfaces. We cut the link complement in S 3 along the state surface, and then describe how to decompose the result into a collection of topological
- PDF / 504,285 Bytes
- 17 Pages / 439.36 x 666.15 pts Page_size
- 94 Downloads / 223 Views
Decomposition into 3-Balls
In this chapter, we start with a connected link diagram and explain how to construct state graphs and state surfaces. We cut the link complement in S 3 along the state surface, and then describe how to decompose the result into a collection of topological balls whose boundaries have a checkerboard coloring. There are two steps to this decomposition; the first is explained in Sect. 2.2, and the second in Sect. 2.3. Finally, in Sect. 2.4, we briefly describe how to generalize the decomposition to a broader class of links considered by Ozawa in [76]. The combinatorics of the decomposition will be used heavily in later chapters to prove our results. Consequently, in this chapter we will define terminology that will allow us to refer to these combinatorial properties efficiently. Thus the terminology and results of this chapter are important for all the following chapters.
2.1 State Circles and State Surfaces Let D be a connected link diagram, and x a crossing of D. Recall that associated to D and x are two link diagrams, each with one fewer crossing than D, called the A-resolution and B-resolution of the crossing. See Fig. 1.1 on p. 4. A state of D is a choice of A- or B-resolution for each crossing. Applying a state to the diagram, we obtain a crossing free diagram consisting of a disjoint collection of simple closed curves on the projection plane. We call these curves state circles. The all-A state of the diagram D chooses the A-resolution at each crossing. We denote the union of corresponding state circles by sA .D/, or simply sA . Similarly, one can define an all-B state and state circles sB D sB .D/. Start with the all-A state of a diagram. From this, we may form a connected graph in the plane. Definition 2.1. Let sA be the union of state circles in the all-A state of a diagram D. To this union of circles, we attach one edge for each crossing, which records the location of the crossing. (These edges are dashed in Fig. 1.1 on p. 4.) The resulting D. Futer et al., Guts of Surfaces and the Colored Jones Polynomial, Lecture Notes in Mathematics 2069, DOI 10.1007/978-3-642-33302-6 2, © Springer-Verlag Berlin Heidelberg 2013
17
18
2 Decomposition into 3-Balls
Fig. 2.1 Left to right: A diagram. The graph HA . The state surface SA
graph is trivalent, with edges coming from crossings of the original diagram and from state circles. To distinguish between these two, we will refer to edges coming from state circles just as state circles, and edges from crossings as segments. This graph will be important in the arguments below. We will call it the graph of the A-resolution, and denote it by HA . In the introduction, we introduced the A-state graph GA . This will factor into our calculations in later chapters. For now, note that GA is obtained from HA by collapsing state circles to single vertices. We may similarly define the graph of the B-resolution, HB , and the B-state graph GB . Indeed, every construction that follows will work with only minor modifications (involving handedness) if we re
Data Loading...
