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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]

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]

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]

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]

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]

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]