A new invariant of planar knotoids and finite-type invariants

Abstract

In this paper, we introduce a new polynomial invariant of planar (but not spherical) knotoids, which we call the winding signed sum polynomial. This Laurent polynomial invariant of planar knotoids denoted by [Formula: see text] is a type-one Vassiliev invariant. This invariant might tell whether a planar knotoid is a zero-height or nonzero-height planar knotoid. It also might distinguish between a planar knotoid and its inverse, while the affine index polynomial defined by Gügümcü and Kauffman in [New invariants of knotoids, Eur. J. Combin. 65 (2017) 186–229] (that is also a type-one Vassiliev invariant) cannot distinguish between a planar knotoid and its inverse. We also define some geometric invariants of planar knotoids, and give lower bounds for these invariants using the winding signed sum polynomial, which helps in computing these geometric invariants that are easy to define, but hard to calculate.