Recent developments in knot theory provide a method for computing the crossover number for special types of knots and links ($\lbrack$K1$\rbrack$,$\lbrack$LT$\rbrack$,$\lbrack$MuT2$\rbrack$). With this information, questions involving the asymptotic behavior of knots with a fixed crossover number (as the crossover number goes to infinity) can be addressed. An exact count of 4-plat knots and links is obtained, thus proving that the number of prime knots grows at least exponentially. Further, a lower bound of the number of Montesinos knots is produced and some special classes of 4-plats are counted. Many of these results have appeared in $\lbrack$ES1$\rbrack$. / A knot or (2-component) link L can be factored (non-uniqely) into the sum of two 2-string tangles, say A and B. We write L = A + B. Given a system of equations of this kind, some of the knots and tangles involved are treated as known quantities, others as unknown quantities. We want to solve the system for the unknowns. If all tangles involved are rational and all knots and links are 4-plats, we can always find all possible solutions. This is called rational tangle calculus. In more general equations, some partial answers are obtained. The main techniques are the theory of two-fold branched covering spaces, Dehn surgery, and the classification of certain 3-manifolds (Lens spaces and Seifert fiber spaces). This tangle calculus is applied to a model for site-specific DNA recombination. Most of the results involving tangle calculus will appear in $\lbrack$ES2$\rbrack$. / In the last chapter I compile a table of all arborescent tangles with less than 6 crossings in a minimal projection. The chirality of these tangles is determined. / Source: Dissertation Abstracts International, Volume: 49-10, Section: B, page: 4348. / Major Professor: De Witt Sumners. / Thesis (Ph.D.)--The Florida State University, 1988.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_77870 |
| Contributors | Ernst, Claus., Florida State University |
| Source Sets | Florida State University |
| Language | English |
| Detected Language | English |
| Type | Text |
| Format | 130 p. |
| Rights | On campus use only. |
| Relation | Dissertation Abstracts International |
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