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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]

Doodles were introduced in [R. Fenn and P. Taylor, Introducing doodles, in Topology of Low-Dimensional Manifolds, Lecture Notes in Mathematics, Vol. 722 (Springer, Berlin, 1979), pp. 37–43] but were restricted to embedded circles in the 2 -sphere. Khovanov [M. Khovanov, Doodle groups, Trans. Amer. Math. Soc.349 (1997) 2297–2315] extended the idea to immersed circles in the 2 -sphere. In this paper, we further extend the range of doodles to any closed oriented surface. Uniqueness of minimal representatives is proved, and various examples of doodles are given with their minimal representatives. We also introduce the notion of virtual doodles, and show that there is a natural one-to-one correspondence between doodles on surfaces and virtual doodles on the plane. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the (co)homology theory of biquandles, derived biquandle cocycle invariants for oriented surface-links using broken surface diagrams and how to compute the biquandle cocycle invariants from marked graph diagrams. We also develop the shadow (co)homology theory of biquandles and construct the shadow biquandle cocycle invariants for oriented surface-links. [ABSTRACT FROM AUTHOR]

We discuss Gauss codes of virtual diagrams and virtual doodles. The notion of a left canonical Gauss code is introduced and it is shown that oriented virtual doodles are uniquely presented by left canonical Gauss codes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the (co)homology group of a multiple conjugation biquandle. It is the (co)homology group of the prismatic chain complex, which is related to the homology of foams introduced by J. S. Carter, modulo a certain subchain complex. We construct invariants for S 1 -oriented handlebody-links using 2 -cocycles. When a multiple conjugation biquandle X × ℤ type X Y is obtained from a biquandle X using n -parallel operations, we provide a 2 -cocycle (or 3 -cocycle) of the multiple conjugation biquandle X × ℤ type X Y from a 2 -cocycle (or 3 -cocycle) of the biquandle X equipped with an X -set Y. [ABSTRACT FROM AUTHOR]

It is well known that surface-links in 4 -space can be presented by diagrams on the plane of 4 -valent spatial graphs with makers on the vertices, called marked graph diagrams. In this paper, we extend the method of presenting surface-links by marked graph diagrams to presenting immersed surface-links. We also give some moves on marked graph diagrams that preserve the ambient isotopy classes of their presenting immersed surface-links. [ABSTRACT FROM AUTHOR]

Twisted links are a generalization of virtual links. As virtual links correspond to abstract links on orientable surfaces, twisted links correspond to abstract links on (possibly non-orientable) surfaces. In this paper, we introduce the notion of the double covering of a twisted link. It is defined by considering the orientation double covering of an abstract link or alternatively by constructing a diagram called a double covering diagram. We also discuss links in thickened surfaces, their diagrams and their stable equivalence classes. Bourgoin's twisted knot group is understood as the virtual knot group of the double covering. [ABSTRACT FROM AUTHOR]

By using the cohomology theory of quandles, quandle cocycle invariants and shadow quandle cocycle invariants are defined for oriented links and surface-links via broken surface diagrams. By using symmetric quandles, symmetric quandle cocycle invariants are also defined for unoriented links and surface-links via broken surface diagrams. A marked graph diagram is a link diagram possibly with 4-valent vertices equipped with markers. Lomonaco, Jr. and Yoshikawa introduced a method of describing surface-links by using marked graph diagrams. In this paper, we give interpretations of these quandle cocycle invariants in terms of marked graph diagrams, and introduce a method of computing them from marked graph diagrams. [ABSTRACT FROM AUTHOR]

A. S. Lipson constructed two state models yielding the same classical link invariant obtained from the Kauffman polynomial F(a, u). In this paper, we apply Lipson's state models to marked graph diagrams of surface-links, and observe when they induce surface-link invariants. [ABSTRACT FROM AUTHOR]

Quandle cocycle invariants form a powerful and well-developed tool in knot theory. This paper treats their variations - namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular cases of the latter. As an application, several constructions from the shadow world are extended to the positive and twisted cases. Another application is a sharpening of twisted quandle cocycle invariants for multi-component links. [ABSTRACT FROM AUTHOR]

We introduce two kinds of structures, called v-structures and t-structures, on biquandles. These structures are used for colorings of diagrams of virtual links and twisted links such that the numbers of colorings are invariants. Given a biquandle or a quandle, we give a method of constructing a biquandle with these structures. Using the numbers of colorings, we show that Bourgoin's twofoil and non-orientable virtual m-foils do not represent virtual links [ABSTRACT FROM AUTHOR]

We study bridge presentations of virtual knots, and determine the virtual bridge numbers of pseudo-prime virtual knots with real crossing numbers less than 5, except two virtual knots. [ABSTRACT FROM AUTHOR]

We introduce the virtual magnetic Kauffman bracket skein module, and show how to find a skein relation for the f-polynomial through the module. Furthermore, we give a unified approach to constructions of skein relations which hold under certain conditions. [ABSTRACT FROM AUTHOR]

We introduce an equivalence relation, called stable equivalence, on knot diagrams and closed generically immersed curves on surfaces. We give bijections between the set of abstract knots, the set of virtual knots, and the set of the stable equivalence classes of knot diagrams on surfaces. Using these bijections, we define concordance and link homology for virtual links. As an application, it is shown that Kauffman's example of a virtual knot diagram is not equivalent to a classical knot diagram. [ABSTRACT FROM AUTHOR]

Homomorphisms on quandle cohomology groups that raise the dimensions by one are studied in relation to the cocycle state-sum invariants of knots and knotted surfaces. Skein relations are also studied. [ABSTRACT FROM AUTHOR]

Geometric representations of cycles in quandle homology theory are given in terms of colored knot diagrams. Abstract knot diagrams are generalized to diagrams with exceptional points which, when colored, correspond to degenerate cycles. Bounding chains are realized, and used to obtain equivalence moves for homologous cycles. The methods are applied to prove that boundary homomorphisms in a homology exact sequence vanish. [ABSTRACT FROM AUTHOR]

The notion of an abstract link diagram is re-introduced with a relationship with Kauffman's virtual knot theory. It is proved that there is a bijection from the equivalence classes of virtual link diagrams to those of abstract link diagrams. Using abstract link diagrams, we have a geometric interpretation of the group and the quandle of a virtual knot. A generalization to higher dimensional cases is introduced, and the state-sum invariants are treated. [ABSTRACT FROM AUTHOR]