In this thesis, we show that every periodic virtual knot can be realized as the closure of a periodic virtual braid. If K is a q-periodic virtual knot with quotient K_*, then the knot group G_{K_*} is a quotient of G_K and we derive an explicit q-symmetric Wirtinger presentation for G_K, whose quotient is a Wirtinger presentation for G_{K_*}. When K is an almost classical knot and q=p^r, a prime power, we show that K_* is also almost classical, and we establish a Murasugi-like congruence relating their Alexander polynomials modulo p.
This result is applied to the problem of determining the possible periods of a virtual knot $K$. For example, if K is an almost classical knot with nontrivial Alexander polynomial, our result shows that K can be p-periodic for only finitely many primes p. Using parity and Manturov projection, we are able to apply the result and derive conditions that a general q-periodic virtual knot must satisfy. The thesis includes a table of almost classical knots up to 6 crossings, their Alexander polynomials, and all known and excluded periods. / Thesis / Doctor of Philosophy (PhD)
Identifer | oai:union.ndltd.org:mcmaster.ca/oai:macsphere.mcmaster.ca:11375/21006 |
Date | January 2017 |
Creators | White, Lindsay |
Contributors | Boden, Hans U., Nicas, Andrew J., Mathematics |
Source Sets | McMaster University |
Language | English |
Detected Language | English |
Type | Thesis |
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