Alexander invariants of periodic virtual knots

Hans U. Boden, Andrew J. Nicas, Lindsay White
2018 Dissertationes Mathematicae  
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
more » ... ing 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. iii
doi:10.4064/dm785-3-2018 fatcat:rlfpfp4odrd3flw5vf2oacvbda