The foundation of Luecke’s course M: 396 Geometry and Topology is that collaboration amongst mathematicians and biologists caused tremendous gains in DNA research. The field of topology has led to significant strides in understanding of the topological properties of the genetic molecule DNA. Through the integration of biological phenomena and knowledge of topology and Euclidean geometry, biologists can describe and quantize enzyme mechanisms and therefore determine enzyme mechanisms causing the changes. Understanding mathematical applications in contexts outside of mathematics on any level helps to explain why mathematics is a core content area in primary and secondary education. Requiring secondary educators to take such a course could result in mathematics taught with real world application on the secondary level as well as on the graduate level, as shown in Luecke’s course. / text

This thesis investigates the topology and geometry of hyperbolic knot complements that are commensurable with other knot complements. In chapter 3, we provide an infinite family examples of hyperbolic knot complements commensurable with exactly two other knot complements. In chapter 4, we exhibit an obstruction to knot complements admitting exceptional surgeries in conjunction with hidden symmetries. Finally, in chapter 5, we discuss the role of surfaces embedded in 3-orbifolds as it relates to hidden symmetries. / text

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Gossypium hirsutum, upland cotton, is one of the major crops grown in the United States and the world. Upland cotton is cultivated in areas that are ideal breeding grounds for the difficult to manage, southern root-knot nematode (RKN), Meloidogyne incognita. Host plant resistance is the most effective way to control RKN populations. However, resistance used in most breeding programs stems from a few related sources. Novel sources of resistance have been identified but have yet to be introduced into elite breeding lines or genetically studied. The objectives of this study are two-fold. The first is to develop elite germplasm by introgressing RKN resistance from primitive accessions into modern cotton genotypes via backcrossing. The second is to use simple sequence repeats (SSRs) to identify loci associated with RKN resistance in the primitive accessions. The genotypes used will be: 1) inoculated with M. incognita, 2) phenotypically analyzed by measuring the nematode reproduction as eggs per gram of fresh root and host response using a root gall index, 3) genetically evaluated by using SSR markers to detect polymorphisms between the RKN resistant TX accessions and DP90 (susceptible genotype), and 4) analyzed using linkage and mapping software. Elite germplasm that contains: 1) high yield potential and a high level of RKN- resistance or 2) high fiber quality and RKN-resistance was developed by performing two backcrosses based on phenotypic analyses. A third screen is currently underway to ensure the introgression of the RKN resistance genes. Agronomic tests will need to be done before the germplasm is released. Genetic analyses using SSR-based primer sets of the TX accessions did not yield expected results. Of the 508 primers sets tested, only 31 were polymorphic between the TX accessions and DP90. A bulked segregant analysis approach was used to test the 31 primer sets on the resistant and susceptible bulks of the F2 population but no polymorphisms were seen. However, analyses found that the TX accessions were more genetically similar to Mexico Wild Jack Jones than to Clevewilt 6-3-5. More work needs to be done to understand the mechanism of RKN resistance in the TX accessions.

Quandles were introduced to Knot Theory in the 1980s as an almost complete algebraic invariant for knots and links. Like their more basic siblings, groups, they are difficult to distinguish so a major challenge is to devise means for determining when two quandles having different presentations are really different. This thesis addresses this point by studying algebraic aspects of quandles. Following what is mainly a recapitulation of existing work on quandles, we firstly investigate how a link quandle is related to the quandles of the individual components of the link. Next we investigate coset quandles. These are motivated by the transitive action of the operator, associated and automorphism group actions on a given quandle, allowing techniques of permutation group theory to be used. We will show that the class of all coset quandles includes the class of all Alexander quandles; indeed all group quandles. Coset quandles are used in two ways: to give representations of connected quandles, which include knot quandles; and to provide target quandles for homomorphism invariants which may be useful in enabling one to distinguish quandles by counting homomorphisms onto target quandles. Following an investigation of the information loss in going from the fundamental quandle of a link to the fundamental group, we apply our techniques to calculations for the figure eight knot and braid index two knots and involving lower triangular matrix groups. The thesis is rounded out by two appendices, one giving a short table of knot quandles for knots up to six crossings and the other a computer program for computing the homomorphism invariants.

Quandles were introduced to Knot Theory in the 1980s as an almost complete algebraic invariant for knots and links. Like their more basic siblings, groups, they are difficult to distinguish so a major challenge is to devise means for determining when two quandles having different presentations are really different. This thesis addresses this point by studying algebraic aspects of quandles. Following what is mainly a recapitulation of existing work on quandles, we firstly investigate how a link quandle is related to the quandles of the individual components of the link. Next we investigate coset quandles. These are motivated by the transitive action of the operator, associated and automorphism group actions on a given quandle, allowing techniques of permutation group theory to be used. We will show that the class of all coset quandles includes the class of all Alexander quandles; indeed all group quandles. Coset quandles are used in two ways: to give representations of connected quandles, which include knot quandles; and to provide target quandles for homomorphism invariants which may be useful in enabling one to distinguish quandles by counting homomorphisms onto target quandles. Following an investigation of the information loss in going from the fundamental quandle of a link to the fundamental group, we apply our techniques to calculations for the figure eight knot and braid index two knots and involving lower triangular matrix groups. The thesis is rounded out by two appendices, one giving a short table of knot quandles for knots up to six crossings and the other a computer program for computing the homomorphism invariants.

Quandles were introduced to Knot Theory in the 1980s as an almost complete algebraic invariant for knots and links. Like their more basic siblings, groups, they are difficult to distinguish so a major challenge is to devise means for determining when two quandles having different presentations are really different. This thesis addresses this point by studying algebraic aspects of quandles. Following what is mainly a recapitulation of existing work on quandles, we firstly investigate how a link quandle is related to the quandles of the individual components of the link. Next we investigate coset quandles. These are motivated by the transitive action of the operator, associated and automorphism group actions on a given quandle, allowing techniques of permutation group theory to be used. We will show that the class of all coset quandles includes the class of all Alexander quandles; indeed all group quandles. Coset quandles are used in two ways: to give representations of connected quandles, which include knot quandles; and to provide target quandles for homomorphism invariants which may be useful in enabling one to distinguish quandles by counting homomorphisms onto target quandles. Following an investigation of the information loss in going from the fundamental quandle of a link to the fundamental group, we apply our techniques to calculations for the figure eight knot and braid index two knots and involving lower triangular matrix groups. The thesis is rounded out by two appendices, one giving a short table of knot quandles for knots up to six crossings and the other a computer program for computing the homomorphism invariants.