Homological invariants of strongly invertible knots

QA Mathematics

Snape, Michael ; Engineering and Physical Sciences Research Council (EPSRC) ; Watson, Liam

University:University of Glasgow

Department:School of Mathematics and Statistics

关键词: Knot theory, low-dimensional topology.;   

Others  :  http://theses.gla.ac.uk/39015/1/2018SnapePhD.pdf

来源: University of Glasgow

【 摘 要 】

This thesis explores the relationship between Khovanov homology and strongly invertible knots through the use of a geometric construction due to Sakuma. On the one hand, new homological and polynomial invariants of strongly invertible knots are extracted from Sakuma's construction, all of which are related to Khovanov homology. Conversely, these invariants are used to study the two-component links and annular knots obtained from Sakuma's construction, the latter of which are almost entirely disjoint from the class of braid closures. Applications include the problem of unknot detection in the strongly invertible setting, the efficiency of an invariant when compared with the $\eta$-polynomial of Kojima and Yamasaki, and the use of polynomial invariants to bound the size of the intrinsic symmetry group of a two-component Sakuma link. We also define a new quantity, $\varkappa_A$, and conjecture that it is an invariant of strongly invertible knots.