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Twisted links are a generalization of classical links and correspond to stably equivalence classes of links in thickened surfaces. In this paper, we introduce twisted intersection colorings of a diagram and construct two invariants of a twisted link using such colorings. As an application, we show that there exist infinitely many pairs of twisted links such that for each pair the two twisted links are not equivalent but their double coverings are equivalent. We also introduce a method of constructing a pair of twisted links whose double coverings are equivalent. [ABSTRACT FROM AUTHOR]

A virtual link is a generalization of a classical link that is defined as an equivalence class of certain diagrams, called virtual link diagrams. It is further generalized to a twisted link. Twisted links are in one-to-one correspondence with stable equivalence classes of links in oriented thickenings of (possibly nonorientable) closed surfaces. By definition, equivalent virtual links are also equivalent as twisted links. In this paper, we discuss when two virtual links are equivalent as twisted links, and give a necessary and sufficient condition for this to be the case. [ABSTRACT FROM AUTHOR]

In this paper we compute a presentation for the group of ring motions of the split union of a Hopf link with Euclidean components and a Euclidean circle. A key part of this work is the study of a short exact sequence of groups of ring motions of general ring links in R 3. This sequence allowed us to build the main result from the previously known case of the ring group with one component, which a particular case of the ring groups studied by Brendle and Hatcher. This work is a first step towards the computation of a presentation for groups of motions of H-trivial links with an arbitrary number of components. [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]

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]

It is known that every surface-link can be presented by a marked graph diagram, and such a diagram presentation is unique up to moves called Yoshikawa moves. G. Kuperberg introduced a regular isotopy invariant, called the quantum A 2 invariant, for tangled trivalent graph diagrams. In this paper, a polynomial for a marked graph diagram is defined by use of the quantum A 2 invariant and it is studied how the polynomial changes under Yoshikawa moves. The notion of a ribbon marked graph is introduced to show that this polynomial is useful for an invariant of a ribbon 2-knot. [ABSTRACT FROM AUTHOR]

We introduce the notion of a ribbon-clasp surface-link, which is a generalization of a ribbon surface-link. We generalize the notion of a normal form on embedded surface-links to the case of immersed surface-links and prove that any immersed surface-link can be described in a normal form. It is known that an embedded surface-link is a ribbon surface-link if and only if it can be described in a symmetric normal form. We prove that an immersed surface-link is a ribbon-clasp surface-link if and only if it can be described in a symmetric normal form. We also introduce the notion of a ribbon-clasp normal form, which is a simpler version of a symmetric normal form. [ABSTRACT FROM AUTHOR]

Braided surfaces are surfaces embedded or immersed in the bidisk D 2 × D 2 which are projected onto the second factor of the bidisk as branched covering maps. A simple and embedded braided surface is described by a graph in the 2-disk, called a chart. We generalize the chart description method so that one can consider braided surfaces which are not necessarily simple or embedded. We show that if a braided surface is regular, then it can be described by a chart called regular, and that such a chart is unique up to regular chart move equivalence. [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]

Chart descriptions are a graphic method to describe monodromy representations of various topological objects. Here we introduce a chart description for hyperelliptic Lefschetz fibrations, and show that any hyperelliptic Lefschetz fibration can be stabilized by fiber-sum with certain basic Lefschetz fibrations. [ABSTRACT FROM AUTHOR]

The notion of a braid is generalized into two and three dimensions. Two-dimensional braids are described by braid monodromies or graphics called charts. In this paper we introduce the notion of curtains, and show that three-dimensional braids are described by braid monodromies or curtains. [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]

Abstract: Chart descriptions are a graphic method to describe monodromy representations of various topological objects. Here we introduce a chart description for genus-two Lefschetz fibrations, and show that any genus-two Lefschetz fibration can be stabilized by fiber-sum with certain basic Lefschetz fibrations. [Copyright &y& Elsevier]

Abstract: A quandle is a set with a binary operation satisfying certain conditions related to Reidemeister moves in knot theory. First we give an example of a quandle with subsets which are not subquandles but closed under the quandle operation. We introduce a method to produce a quandle from an invertible dynamical system. Our example is generalized to such dynamical quandles. [Copyright &y& Elsevier]

Abstract: Various topological objects; 2-dimensional braids, braided surfaces, Lefschetz fibrations of 4-manifolds, algebraic curves, etc., can be treated by their monodromy representations. We introduce a graphical method, called the chart description method, to describe and calculate such monodromy representations. This method is, in a sense, a generalization of the picture method. [Copyright &y& Elsevier]

A quandle is a set with a self-distributive binary operation satisfying a certain condition. Here we construct a monoid (a semi-group with the identity) associated with a quandle. This monoid has a structure of a quandle, which contains the original quandle as a sub-quandle. We call it the enveloping monoidal quandle. The purpose of this paper is to introduce the notion of the enveloping monoidal quandle, and to investigate it. [Copyright &y& Elsevier]

In this paper a purely algebraic condition for a word in a free group to be representable by a simple curve on a punctured plane will be given.As an application, an algorithm for simple closed curves on a punctured plane will be obtained. Our solution is different from any algorithm due to Reinhart [Ann. of Math. 75 (1962) 209], Zieschang [Math. Scand. 17 (1965) 17] or Chillingworth [Bull. London Math. Soc. 1 (1969) 310]. Although the study here will be confined to the case of a plane, similar arguments could be carried out on the 2-sphere. This research was motivated by monodromy problems appearing in Lefschetz fibrations and surface braids. See [Math. Proc. Cambridge Philos. Soc. 120 (1996) 237; Kamada, Braid and Knots Theory in Dimension Four, American Mathematical Society, 2002; Kamada and Matsumoto, in: Proceedings of the International Conference on Knot Theory “Knots in Hellas ''98”, World Scientific, 2000, p. 118; Kamada and Matsumoto, Enveloping monoidal quandles, Preprint, 2002; Matsumoto, in: S. Kojima et al. (Eds.), Proc. the 37th Taniguchi Sympos., World Scientific, 1996, p. 123]. [Copyright &y& Elsevier]

Vassiliev''s 1-knot invariants are defined in a combinatorial way as finite type invariants. The notion of a finite type invariant associated with given local operations is defined. In this paper we prove that a local operation called a 1-handle surgery is an unknotting operation for generically immersed surfaces in 4-space and investigate the finite type invariants of such surfaces in 4-space associated with this operation. [Copyright &y& Elsevier]

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]

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]

A quandle is an algebra with two binary operations satisfying three conditions which are related to Reidemeister moves in knot theory. In this paper we introduce the notion of the (canonical) tensor product of a quandle. The tensor product of the knot quandle or the knot symmetric quandle of a surface-link in 4-space can be used to classify or construct invariants of 1-handles attaching to the surface-link. We also compute the tensor products for dihedral quandles and their symmetric doubles. [ABSTRACT FROM AUTHOR]

Examines the representation of link types in R3 as a closed braid by Alexander's and Markov's theorems. Correlation between representations through Markov moves; Generalization of braid concept within four-dimensional space; Analogy to theorems.

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]

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]

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]

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]

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]

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]

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]

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]

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]