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A doodle is a collection of immersed circles without triple intersections in the 2-sphere. It was shown by the second author and Tayler that doodles induce commutator identities (identities amongst commutators) in a free group. In this paper, we observe this idea more closely by concentrating on doodles with proper noose systems and elementary commutator identities. In particular, we show that there is a bijection between cobordism classes of colored doodles and weak equivalence classes of elementary commutator identities. [ABSTRACT FROM AUTHOR]

Abstract: We introduce a two-variable polynomial invariant of a long virtual knot, which dominates the Kauffman f-polynomial and the Miyazawa polynomial of the closure. Our invariant satisfies a product formula for the concatenation product of long virtual knots. It describes a formula of the Miyazawa polynomial of a ‘connected sum’ of two virtual knots. It also gives lower bounds for the real crossing number and the virtual crossing number of a long virtual knot. [Copyright &y& Elsevier]

Biquandles and multiple conjugation biquandles are algebras which are related to links and handlebody-links in 3-space. Cocycles of biquandles and multiple conjugation quandles can be used to construct state-sum type invariants of links and handlebody-links. In this paper we discuss cocycles of a certain class of biquandles and multiple conjugation biquandles, which we call G -Alexander biquandles and G -Alexander multiple conjugation biquandles, with a relationship with group cocycles. We give a method to obtain a (biquandle or multiple conjugation biquandle) cocycle from a group cocycle. [ABSTRACT FROM AUTHOR]

A virtual doodle is an equivalence class of virtual diagrams under an equivalence relation generated by flat version of classical Reidemeister moves and virtual Reidemeister moves such that Reidemeister moves of type 3 are forbidden. In this paper we discuss colorings of virtual diagrams using an algebra, called a doodle switch, and define an invariant of virtual doodles. Besides usual colorings of diagrams, we also introduce doubled colorings. [ABSTRACT FROM AUTHOR]