59 research outputs found
New Quantum Invariants of Planar Knotoids
In this paper we discuss the applications of knotoids to modelling knots in
open curves and produce new knotoid invariants. We show how invariants of
knotoids generally give rise to well-behaved measures of how much an open curve
is knotted. We define biframed planar knotoids, and construct new invariants of
these objects that can be computed in polynomial time. As an application of
these invariants we improve the classification of planar knotoids with up to
five crossings by distinguishing several pairs of prime knotoids that were
conjectured to be distinct by Goundaroulis et al.Comment: 29 pages, 21 figures, comments are welcom
Knotoids
We introduce and study knotoids. Knotoids are represented by diagrams in a
surface which differ from the usual knot diagrams in that the underlying curve
is a segment rather than a circle. Knotoid diagrams are considered up to
Reidemeister moves applied away from the endpoints of the underlying segment.
We show that knotoids in generalize knots in and study the
semigroup of knotoids. We also discuss applications to knots and invariants of
knotoids.Comment: 27 pages, 7 figures. References update
Graphoids
We study invariants of virtual graphoids, which are virtual spatial graph
diagrams with two distinguished degree-one vertices modulo graph Reidemeister
moves applied away from the distinguished vertices. Generalizing previously
known results, we give topological interpretations of graphoids. There are
several applications to virtual graphoid theory. First, virtual graphoids are
suitable objects for studying knotted graphs with open ends arising in
proteins. Second, a virtual graphoid can be thought of as a way to represent a
virtual spatial graph without using as many crossings, which can be
advantageous for computing invariants
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