Hyperbolic 3-Manifolds Dehn Surgery Approach to the Figure-Eight Knot

Home , Dehn surgery, Hyperbolic link, Satellite knot, Twist knot, Whitehead link

Hyperbolic 3-Manifolds

Dehn Surgery Approach to The Figure-Eight

Knot Complement

Sarah Dubicki, ’11 Math 490: Senior Independent Study Advisor Dr. Barbara Nimershiem

May 5, 2011

Abstract

This project provides an examination of knot complements and their rela- tionship to hyperbolic 3-manifolds. It begins with the study of knot theory, specifically knot complements in S3. The knots we will be focusing on are those whose complements are hyperbolic, which includes a class of knots known as the twist knots, and in particular we will look at the hyperbolic structure of the link complement for the first even twist knot, the figure-eight knot. We then take the Dehn surgery approach to obtaining the structure of the figure-eight knot complement. By performing Dehn surgery on the Whitehead link, we are able to obtain the figure-eight knot. This project concludes by examining the

Dehn surgery on the Whitehead link that yields the ﬁgure-eight knot. The per- formance of Dehn surgery on the hyperbolic structure of the Whitehead link leads us to the hyperbolic structure of the ﬁgure-eight knot complement.

1 I. Background Information

We begin in knot theory with the deﬁnition of a knot. We say that a knot is a closed curve in S3 that is nowhere self-intersecting. In knot theory, there exist three categories of knots: torus knots, satellite knots, and hyperbolic knots. A

(p,q)-torus knot is a closed loop around an unknotted torus with p meridians and q longitudes, where p and q are relatively prime. A satellite knot is essen- tially a knot inside a knotted torus, and a hyperbolic knot is a knot such that its complement in S3 has a hyperbolic structure. These categories are deﬁned such that every knot falls into exactly one category, as seen in [2] p 119. We are interested in the hyperbolic category of knots. A subcategory of the hyperbolic knots is the class of twist knots with the trefoil knot removed. A twist knot is a twisted closed loop with any number of half-twists added.

Figure 1: A closed loop with no twists added, from [12].

Figure 2: The ﬁgure-eight knot (the second twist knot): a closed loop with two half twists added, from [11].

2 This paper will also involve work with the Whitehead Link. The standard projection of this link is as shown in Figure 3. However, this paper will use the projection as shown in Figure 4 for the link, achieved through a series of identity-preserving Reidemeister moves.

Figure 3: The standard projection of the Whitehead Link.

Figure 4: The alternate projection of the Whitehead Link.

II. Twist Knot Complement - Adams’s Approach

In the ﬁrst chapter of his Ph.D. dissertation [1], entitled Explicit Hyper- bolic Structures on Link complements, Adams provides an algorithm that can be applied to a hyperbolic link in S3 to yield the hyperbolic structure of its complement (in S3). Adams’s method begins by putting a link, L, with n crossings in a regular projection where L lies on the projection except at a crossing.

At a crossing, L lies on the surface of a sphere, which we call Si, where i is the

3 crossing number and 1 ≤ i ≤ n. The 3-cell in S − L bounded by Si is Bi. As

3 such, Adams thinks of the complement of L in S3 in terms of two subsets of S3:

n C1, which is the half space above the projection plane with L and ∪i=1(Int(Bi)) removed, and C2, which is the half space below the projection plane with L

n and ∪i=1(Int(Bi)) removed. The union of C1 and C2, C1 ∪ C2, is the space 3 n 3 {S − L − ∪i=1(Int(Bi))}. This is nearly the complement of L in S , we need to ﬁnd a way to ﬁll in the missing Bi’s.

Adams notes that Bi is homeomorphic to an octahedron, which he calls Oi.

He then deﬁnes four disks in Oi with which we determine four 3-cells in Oi, each bounded by two disks and two faces of the octahedron. Each of these 3-cells shares the vertical line segment from the overstrand to the understrand. C1 shares 4 faces of Oi each with two of the 3-cells, and C2 shares 4 faces of Oi

0 each with the two remaining 3-cells. We obtain a new space C1 by gluing the

0 ﬁrst two 3-cells along the four faces they share with C1 and C2 by gluing the remaining two 3-cells along the four faces they share with C2. Identifying the

0 0 boundaries of C1 and C2 using previous identiﬁcations ﬁlls in the missing Bi’s and gives us a space homeomorphic to S3 − L.

We can think of this process as pinching the top 3-cell around the overstrand so it touches itself along the edge from the overstrand to the understrand and pinching the bottom 3-cell around the understrand so that it touches itself in the same edge. This process deforms our link projection so that when it is

ﬂattened to a plane, it becomes a directed graph which corresponds to a poly-

0 0 hedron. The boundary of C1 and the boundary C2 of each correspond to a

0 polyhedron-determining directed graph. Flattening C1 to a plane, we have the original link projection whose understrand arcs at a crossing are labeled with the number of that crossing and directed away from the crossing. Flattening

0 C2 to a plane, we have the original link projection whose overstrand arcs at a crossing are labeled with the number of that crossing and directed toward the

4 crossing. Arcs labeled with the same number will be identiﬁed such that arrow directions match in S3 − L, and two edges with the same edge number have the

0 0 same edge type. Our directed graphs divide the boundaries of C1 and C2 into

0 regions that designate faces of our polyhedra. Each face in C1 is identiﬁed to a

0 face in C2 and edge types around both faces must match exactly. Adams also shows that digons (faces with only two edges) may be collapsed to a single edge.

Following this algorithm, we ﬁnd a polyhedron that determines the hyperbolic structure on the complement of L in S3.

III. Figure-Eight Complement - Adams’s Approach

We now look at the explicit case of constructing the complement of the

ﬁgure-eight knot, and begin by laying out two projections of the knot, as in

Figure 5 (one will correspond to the “top” tetrahedron and the other will correspond to the “bottom” tetrahedron).

We will label each crossing, and direct strands at the crossings. For the pro-

Figure 5: Projections of the ﬁgure-eight knot.

jection that corresponds to the “top” tetrahedron, we direct the under-strand at each crossing toward the crossing. For the projection that corresponds to the

“bottom” tetrahedron, we direct the over-strand at each crossing toward the crossing. See Figure 6.

5 We will now designate an orientation on each of the strands, noting that

Figure 6: Directed projections of the ﬁgure-eight knot.

the digons may be collapsed to single lines. Note that there will be two edge types in this case. We will identify edge type 1 with the 1-strand, and we will identify edge type 2 with the 3-strand. This yields the tetrahedra as shown in

Figure 7.

These tetrahedra will be used to determine to the hyperbolic structure of

Figure 7: The two directed graphs above correspond the bottom two tetrahedra with directed edges.

the ﬁgure eight knot complement. We would like to know the angle measures of

6 these tetrahedra, so we will embed them as ideal tetrahedra with one vertex at

∞, one at 0, one at 1, and the last at z, where z ∈ C. A dihedral angle in our tetrahedron is the angle between two faces of our tetrahedron. We label these

i i i angles z1, z2, z3 in a clockwise direction, where i = 1 for the top tetrahedron and i = 2 for the bottom tetrahedron (see Figure 8), noting that opposite dihedral angles are equal, a result in [3] p 345. Further, the transformation f1(z) = z corresponds to putting the vertex on the bottom face of the tetrahedron whose

z−1 angle measure is z1 at the origin, and the transformations f2(z) = z and 1 f3(z) = 1−z correspond to putting the other two vertices at the origin while i i preserving the tetrahedron. Thus, we have zj = fj(z ).

Figure 8: Tetrahedra embedded as ideal tetrahedra.

We know that the sum of dihedral angles about any vertex is π, as explained in [3] p 345. Further, we know that the angle sum around an edge of one of these tetrahedra after all identifications are made must be 2π, as a completeness requirement outlined in Poincaré’sPolyhedron Theorem (for the reader who wishes to see more on this theorem, Maskit provides a well-detailed out- line in section IV of [6]). We can see the completed identifications around two different edges of these tetrahedra in Figure 9

7 Figure 9: The completed identiﬁcation about the edge through z1 and ∞, edge 1, is on the left, and the completed identiﬁcation about the edge through vertex z2 and ∞, edge 2, is on the right.

We have the following system of edge equations

1 2 1 2 1 2 z1 z2 z1 z1 z2 z1 = 1 (1)

1 2 1 2 1 2 z2 z3 z3 z2 z3 z3 = 1 (2)

Here, the first equation corresponds to the identifications about edge 1 and the second equation corresponds to the identifications about edge 2. As an additional requirement from Poincaré’sPolyhedron Theorem, we must have

1 2 1 1 2 z2 z1 z1 = −1. By substitution of z with x and z with y have the following system of completeness requirements.

y − 1 x − 1 x · · x · y · y = 1 (3) y x x − 1 1 1 y − 1 1 y − 1 · · · · · = 1 (4) x 1 − y 1 − x y 1 − x y x − 1 · y · x = −1 (5) x

i π i π Solving this system of equations, we ﬁnd that x = e 3 and y = e 3 . Using these values, we ﬁnd that the angle measure at each of the vertices of both tetrahedra

8 π is 3 . Applying the same algorithm to the Whitehead link in S3, we obtain the polyhedron in Figure 10 for the structure of the Whitehead link complement.

i π This structure is complete if and only if x = ±e 2 .

Figure 10: By performing face identiﬁcations on this polyhedron, we yield the structure of the Whitehead complement in S3. Here, face A is identiﬁed with A0, B with B0, and so on. Image adapted from [8] p. 292

IV. Topological Dehn Surgery

We will now take an alternative approach to the twist-knot complements through a process known as Dehn surgery. The principal idea behind Dehn surgery is that the complement of a solid torus in S3 is a solid torus. We see this concept illustrated in Figure 11.

With Dehn surgery in S3, we can remove a solid torus and then re-identify the removed torus with its complementary torus twisted by p meridian curves and q longitude curves. This Dehn surgery has the integer parameters (p, q) and

9 Figure 11: We think of S3 as two solid spheres with their boundaries identiﬁed. Remove a solid torus by removing a tubular neighborhood about the equator of each sphere. By stretching and identifying, we see that what remains is itself a solid torus. Image from [7] p. 45

10 p is also sometimes identiﬁed by the rational number q . It should be noted that we want p, q relatively prime. It should also be noted that (p, q) and (−p, −q) surgeries are equivalent. The removed torus and its standard meridian and longitude curves are labeled in Figure 12.

Note that if we take our solid torus to be the neighborhood of a component

Figure 12: The removed torus, where µ designates a standard meridian curve and λ designates a standard longitudian curve. Image from [7] p. 45

of our link that we have removed from S3, performing Dehn surgery on this neighborhood will yield our original link, with twists added. In particular, we will take our solid torus to be the neighborhood of the closed simple loop component of Whitehead link and perform Dehn surgery as illustrated in Figure

13 to yield the ﬁgure-eight knot.

V. Hyperbolic Dehn Surgery

We will begin this section by looking at the deﬁnition of completeness as it applies to the hyperbolic knot complement structure. Recall that a metric space, M, is complete if every Cauchy sequence of numbers in M con-

∞ verges to a limit in M. A Cauchy sequence is a sequence {aj}j=0 such that limmin{m,n}→∞ |am − an| = 0. We will ﬁrst explore the concept of completeness as it applies to two-dimensional hyperbolic Dehn ﬁlling and then generalize our

ﬁndings to understand three-dimensional hyperbolic Dehn ﬁlling.

11 Figure 13: We perform −1 surgery (which corresponds to the parameters (1, −1)) on the Whitehead link to obtain the ﬁgure-eight knot.

A zero-dimensional link in two-dimensional space is a collection of 0-spheres, or pairs of points, in a 2-sphere. We will consider the two-dimensional link with

4 components, as pictured in Figure 14. Notice that the complement of this link has the hyperbolic structure pictured in Figure 15, an ideal tetrahedron. To deform the hyperbolic structure, we will start by slicing open one of the cusps.

Re-gluing the cusp by identifying opposite edges straight across as in Figure 16 locally yields the cylinder in Figure 16, which restores the cusp to its original condition, a complete structure.

On the other hand, if we instead re-glue the cusp by identifying opposite edges with a shift, as in Figure 17, we obtain a diﬀerent result. Locally, we obtain the cylinder-like structure in Figure 17. Notice that this space is not complete. As we travel up the cylinder, the distance between edge points approaches 0. In fact, the collection of edge points along line x is a Cauchy

12 Figure 14: Here we have a projection of a 4-component one-dimensional link situated in S2. Image from [10] p. 3.

Figure 15: Here we have the structure of the complement of the link in Figure 14. Notice that this structure is hyperbolic. Image from [10] p. 3.

Figure 16: Slicing open one of the cusps in Figure 15, we yield the top picture. Re-identifying edges straight across (A is identiﬁed with A0 and B is identiﬁed with B0), we yield the bottom picture. Notice that this corresponds to the original cusp, whose structure is complete. Image inspired by [10] p 6.

13 sequence. However, this Cauchy sequence approaches a point on a limiting circle that is not part of our metric space. In order to make this metric space complete, we must ﬁll in the missing limiting circle. This is Dehn ﬁlling in the

2-dimensional case. Deforming all four cusps and performing Dehn ﬁlling in this way yields the hyperbolic structure in Figure 18.

Figure 17: Slicing open one of the cusps in Figure 15, we yield the top picture. Re-identifying edges with a shift (A is identiﬁed with A0 and B is identiﬁed with B0), we yield the bottom picture. Notice that as we travel up the line x, the edge points that we hit make up a Cauchy sequence. However, the limit that the sequence approaches is on a circle that is not a part of our structure. Therefore, this structure is not complete. Image adapted from [10] p 6.

In the 3-dimensional case, a link is a collection of circles in S3, rather than a collection of pairs of points in S3. Whereas the hyperbolic Dehn ﬁlling in the

2-dimensional case is determined by gluing, in the 3-dimensional case gluings between the tetrahedra are rigid, and Dehn ﬁlling is instead determined by the shapes of the tetrahedra. In this case, our cut-open cusps become pyramid-like shapes as in Figure 19. If the cross-sections of this space are parallelograms

14 Figure 18: Deforming each of the cusps in Figure 15 as we have in Figure 17, and ﬁlling in the limiting circles at each cusp, we yield the above complete structure. Image from [10] p. 7.

(as in Figure 19), left, sides will match straight across, ensuring that our structure is complete. However, if cross sections are not parallelograms, sides will match with vertical and horizontal oﬀsets of integer value, as in Figure 19, right.

We will consider the case where the cross-sections are not parallelograms and the vertical and horizontal offsets are relatively prime. Let us consider the example from [10] in Figure 20 where the vertical offset is 5 and the horizontal offset is 2. By slicing the cusp into pieces of fixed height and then identifying the sides of these pieces, we get the cylindrical shape in Figure 21. When we perform the top and bottom identifications, we yield a torus with its central geodesic missing. We must fill in this missing geodesic to complete the structure, this completion being hyperbolic Dehn filling in the 3-dimensional case.

The Dehn ﬁlling parameters are determined by the number of meridian and longitude wraps in this bounding torus, which is determined by the shape of the cusp. In this example, we can look at Figure 21 from above and see, as in

Figure 22, that the Dehn filling coefficients are (2, 5), since the vertical offset is

5 and the horizontal oﬀset is 2.

15 Figure 19: Cutting open the cusp of the structure of a complement of a link in S3, we yield a pyramid-like shape as shown above. If the cross sections of our cut-open cusp are parallelograms, as are those on the left, identifications are made straight across (A is identified with A0) and the resulting structure is complete. However, if cross sections are not parallelograms, sides are identified with a horizontal and vertical offset (B is identified with B0 and w is identified with w0). Image from [10] p. 22.

VI. Figure-Eight Knot Complement: Hyperbolic Dehn Surgery on the Whitehead Link Complement Approach

We will now look at hyperbolic Dehn filling on the hyperbolic structure of the Whitehead link complement, in particular its application in finding the hyperbolic structure of the figure-eight knot complement. Unlike in the previous section, where we began with a certain deformation of a cusp and determined the

Dehn surgery parameters this deformation admitted, in this case, we know the

Dehn surgery coeﬃcients of the surgery on the Whitehead link that yields the

ﬁgure-eight knot, and will use these parameters to determine the cusp deformation that corresponds to this surgery. Recall that the Dehn surgery coeﬃcients in this case are (1, −1), the parameters we will use in this example. This tells us that our bounding torus should have 1 meridian wrap and −1 longitude wrap

(or 1 longitude wrap in the negative direction). Thus, we would like our cusp

16 Figure 20: The cross-sections of this cut-open cusp are such that edges are identified with a vertical offset of 5 and a horizontal offset of 2. Image from [10] p. 23.

Figure 21: By slicing up the ﬁgure in Figure 20 into solid bricks of ﬁxed height and identifying the sides, we yield the above structure. Image from [10] p. 24.

17 Figure 22: A view of Figure 21 from above. Image from [10] p. 24.

identification to have a vertical offset of −1 and horizontal offset of 1. Thus the view from above as in Figure 22 would become as in Figure 23.

Figure 23: In the case of (1, −1) Dehn ﬁlling on the Whitehead link, this is what our view from above (as in Figure 22) of the incomplete cusp would look like.

We now wish to ﬁnd the deformation of the hyperbolic structure of the

Whitehead link complement that will yield the proper corresponding cusp shape.

Recall from section III that the structure of the Whitehead link complement is given by the polyhedron in Figure 10. The value of x determines the shape

18 of the structure, so we wish to ﬁnd the value of x that corresponds to the

Dehn surgery parameters of (1, −1). It should be noted here that, in [8], a non-standard meridian and longitude orientation at a cusp was used, so that

(p, q) Dehn surgery in the standard convention would be (p, −q) Dehn surgery in these papers, which affected some of the formulas given. Performing edge identifications and projecting the polyhedron to the bounding plane, we obtain the two torus-determining polygons in Figure 24 found on page 294 in [8], where the torus on the left is teh bounding torus of the complete cusp, which we will call cusp 1, and the torus on the right is the bounding torus of the incomplete cusp, which we will call cusp 2. From this, using consistency relations, and the fact that cusp 1 is complete, we find formulas for the meridian and longitude parameters, u and v, respectively, of the cusp 2 in terms of x, as outlined in [8].

The formulas are as follows.

u = log x + log(x + 1) − log(x − 1) (6)

v = 4 log x − 2πi (7)

Further, as stated in [8] p 295, the Dehn surgery parameters p, q are determined by the equation pu + qv = 2πi. Solving these equations simultaneously for the parameters px = 1 and qx = 1 (as per the note mentioned earlier), we ﬁnd a √ q √ 1 i 3 1 7 i 3 value for x. The value we ﬁnd is x = − 4 + 4 − 2 2 − 2 .

VII. Comparison of The Two Approaches

According to the Mostow Rigidity Theorem (presented by Meyerhoﬀ on page

48 of [7]), ”if a closed, orientable 3-manifold possesses a hyperbolic structure, then that structure is unique (up to isometry).” The theorem has been extended to include manifolds of dimension greater than 2 with ﬁnite volume. Thus, we would now like to check that the polyhedron that determines the hyperbolic

19 Figure 24: The polygon on the left determines the bounding torus of the complete cusp, and the polygon on the right determines the bounding torus of the 1 incomplete cusp requiring Dehn ﬁlling. Here, x = z, y = w, and y = − x , and 0 x−1 00 x = x and x = 11 − x. From [8] p 294.

structure of the ﬁgure-eight knot complement that we obtained in section III using Adams’s algorithm does in fact the determine the same structure as the polyhedron that we have just obtained in section VI using hyperbolic Dehn surgery. According to Meyerhoﬀ, in [7], volume is a good invariant for hyperbolic

3-manifolds, and all structure-determining polyhedra for a particular hyperbolic

3-manifold have equal volume. However, the volume is not a complete invariant since there do exist distinct hyperbolic 3-manifolds with equal volume.

We will begin by determining the volume of each of the two structure- determining polyhedra for the ﬁgure-eight knot complement. Adams provides a nice formula for the computation of the volume of an ideal tetrahedron based on the three dihedral angle measures in [3], which we will use to compute the volume of our polyhedra. The polyhedron in section III is already split into ideal tetrahedra. Computing the volume of each tetrahedron and adding them together, we ﬁnd 2.02988 ... to be the volume of this polyhedron. The polyhe- √ q √ 1 i 3 1 7 i 3 dron from section VI is Figure 10 with x = − 4 + 4 − 2 2 − 2 . We split

20 this polyhedron into ideal tetrahedra as in Figure 25, and scale these tetrahedra to get the tetrahedra in Figure 26 whose dihedral angles remain the same, but are easier to compute. We compute the volume of each tetrahedron and ﬁnd that the sum of these volumes is 2.02988 ..., the same volume we found for the polyhedron from section III.

Figure 25: The polyhedron in Figure 10 split into tetrahedra, adapted from [8] p 292.

Figure 26: The tetrahedra in Figure 25, scaled to have vertices at 0 and 1.

This should help convince us that the the polyhedra in sections III and VI admit the same structure, but since volume is not a complete invariant, we should perform another check. The projection of the cusp of the polyhedron from section III to the bounding plane gives us a torus-determining polygon, and so the ﬁgure-eight complement has this torus as its boundary. In the polyhedron

21 we found in section VI, we have two cusps, and projecting this polyhedron to the bounding plane will give us two diﬀerent torus-determining polygons. We wish to show that the bounding torus at the completed cusp of this polyhedron

(the cusp that does not require Dehn filling), which corresponds to the figure- eight complement, is in fact the same as the bounding torus of the figure-eight complement. Performing edge identifications and projecting the polyhedron to the bounding plane, we obtain the two torus-determining polygons in Figure

24. The torus on the left bounds the completed cusp of our polyhedron, and the triangulation of this torus is shown in Figure 27. We ﬁnd that m = (1 − x)

2x2+4x−2 and l = 1+x , with our starting point at the origin. We’d like to scale these vectors so that our meridian is 1 and do so by dividing m and l by (1 − x). Our √ q √ 2x2+4x−2 1 i 3 1 7 i 3 longitude becomes 1−x2 . In the case where x = − 4 + 4 − 2 2 − 2 , √ we ﬁnd that the longitude is (3.4641 ... )i, or 2 3i. Using Weeks’s program

SnapPy, we find that the cusp neighborhood of the figure-eight complement is determined by the tiling shown in Figure 28, with the torus-determining polygon boldly outlined. Noticing that the meridian and longitude of this polygon are precisely the meridian and longitude we found for the bounding torus of the complete cusp of the polyhedron in section VI, we can see that this cusp has the same boundary torus as the figure-eight complement.

That both polyhedra have the same, ﬁnite volume and are bound by the same torus at the complete cusp tells us by the Mostow Rigidity Theorem that they determine the same hyperbolic structure.

VIII. Looking Ahead

In general, we may perform Dehn surgery on the Borromean rings (or the

Whitehead link and the sister to the Whitehead link) to yield the class of twist

22 Figure 27: This polygon determines the bounding torus of the complete cusp. From [8] p 296.

Figure 28: Tiling of the cusp neighborhood of the ﬁgure-eight complement with the torus-determining polygon boldly outlined. Each triangle is equilateral with π angles 3 . Image created in SnapPy.

23 knots. Figure 29 illustrates this process for the even twist knots and Figure 30 illustrates this process for the odd twist knots. To get the even twist knots, we ﬁrst perform +1 surgery (which corresponds to the parameters (1, 1)) on the ﬁrst component of the Borromean rings. Note that after this surgery, we yield the alternate projection of the Whitehead link, as illustrated in Figure

4. To yield the nth even twist knot, our next step is to perform −n surgery

(which corresponds to the parameters (−n, 1)) on the Whitehead link. Note that our final projection is a twist knot with 2n crossings, which is in fact the nth even twist knot. Similarly, to obtain the odd twist knots, we first perform −1 surgery (which corresponds to the parameters (−1, 1)) on the first component of the Borromean rings. Note that after this surgery, we yield the mirror image of the alternate projection of the Whitehead link, as illustrated in Figure 4, which we will call the sister to the Whitehead link. To yield the nth odd twist knot, our next step is to perform −(n + 1) surgery (which corresponds to the parameters (−(n + 1), 1)) on the Whitehead link. Note that our final projection is a twist knot with 2n+1 crossings, which is in fact the nth odd twist knot (here we consider the twist knot with 5 crossings the 1st odd twist knot). We can only obtain the even twist knots by performing Dehn surgery on the Whitehead link. We obtain the odd twists knots by performing Dehn surgery on the mirror image of the Whitehead link, or the sister to the Whitehead link.

It would be interesting to take a hyperbolic Dehn surgery approach to ﬁnd- ing the complements of this class of knots and see if patterns emerge.

24 Figure 29: This ﬁgure illustrates the Dehn surgery on the Borromean rings that yields the class of even twist knots.

25 Figure 30: This ﬁgure illustrates the Dehn surgery on the Borromean rings that yields the class of odd twist knots.

26 IX. Acknowledgements

I would ﬁrst like to thank the entire mathematics department at Franklin &

Marshall college for exposing me to a great deal of interesting topics in the ﬁeld and providing a wonderful environment in which I could learn. I would also like to thank my honors committee for their time and dedication, especially Drs.

Michael McCooey and Erik Talvitie for their attention to detail and valuable feedback throughout the process. Lastly, I would like to thank Dr. Barbara

Nimershiem who has advised me through this project and provided an incredible amount of encouragement, support, and invaluable insight.

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