Narrow Results

We construct a new invariant polynomial for long virtual knots. It is a generalization of the Alexander polynomial. We designate it as ζ by meaning an analogy with ζ-polynomial for virtual links. The degree of ζ-polynomial estimates the virtual crossing number. We describe some application of ζ-polynomial for the study of minimal long virtual diagrams with respect to the number of virtual crossings.

We introduce a four-variable index polynomial invariant of long virtual knots that is non-trivial for many long virtual knots, but is trivial for classical knots so that it is an extension of a one-variable index polynomial invariant introduced in [A polynomial invariant of long virtual knots, European J. Combin.30 (2009) 1289–1296]. We give various properties of this polynomial and examples. Also, we use this polynomial invariant for long virtual knots to distinguish virtual knots, and we obtain a polynomial invariant for long flat virtual knots which is very useful to determine whether long flat virtual knots are invertible or not.

We investigate embeddings from the set of long flat virtual knot diagrams to the set of long virtual knot diagrams so that we can construct invariants for long flat virtual knots. Also, we give properties and examples of several invariants for long flat virtual knots via these embeddings and invariants for long virtual knots.