Braid presentation of virtual knots and welded knots

The notion of a virtual knot introduced by L. Kauffman induces the notion of a virtual braid. It is closely related with a welded braid of R. Fenn, R. Rimanyi and C. Rourke. Alexander's and Markov's theorems for virtual knots and braids are proved. Similar results for welded knots and braids are also proved.Comment: 22 pages, 18 figure

Quandles and symmetric quandles for higher dimensional knots

A symmetric quandle is a quandle with a good involution. For a knot in \$R^3\$, a knotted surface in \$R^4\$ or an \$n\$-manifold knot in \$R^{n+2}\$, the knot symmetric quandle is defined. We introduce the notion of a symmetric quandle presentation, and show how to get a presentation of a knot symmetric quandle from a diagram.Comment: 13 pages. It will appear in Banach Center Publication

Knot invariants derived from quandles and racks

The homology and cohomology of quandles and racks are used in knot theory: given a finite quandle and a cocycle, we can construct a knot invariant. This is a quick introductory survey to the invariants of knots derived from quandles and racks.Comment: Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon4/paper8.abs.htm

Cords and 1-handles attached to surface-knots

J. Boyle classified 1-handles attached to surface-knots, that are closed and connected surfaces embedded in the Euclidean 4-space, in the case that the surfaces are oriented and 1-handles are orientable with respect to the orientations of the surfaces. In that case, the equivalence classes of 1-handles correspond to the equivalence classes of cords attached to the surface-knot, and correspond to the double cosets of the peripheral subgroup of the knot group. In this paper, we classify cords and cords with local orientations attached to (possibly non-orientable) surface-knots. And we classify 1-handles attached to surface-knots in the case that the surface-knots are oriented and 1-handles are non-orientable, and in the case that the surface-knots are non-orientable.Comment: It will appear in Boletin de la Sociedad Matematica Mexican

Ribbon-clasp surface-links and normal forms of singular surface-links

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

Chart description for hyperelliptic Lefschetz fibrations and their stabilization

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.Comment: 19 pages, 22 figures. arXiv admin note: substantial text overlap with arXiv:1106.056

Double coverings of twisted links

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

Inverse-free Berlekamp-Massey-Sakata Algorithm and Small Decoders for Algebraic-Geometric Codes

This paper proposes a novel algorithm for finding error-locators of algebraic-geometric codes that can eliminate the division-calculations of finite fields from the Berlekamp-Massey-Sakata algorithm. This inverse-free algorithm provides full performance in correcting a certain class of errors, generic errors, which includes most errors, and can decode codes on algebraic curves without the determination of unknown syndromes. Moreover, we propose three different kinds of architectures that our algorithm can be applied to, and we represent the control operation of shift-registers and switches at each clock-timing with numerical simulations. We estimate the performance in comparison of the total running time and the numbers of multipliers and shift-registers in three architectures with those of the conventional ones for codes on algebraic curves.Comment: 15 pages, submitted to IEEE Transactions on Information Theor

Counting Dirac braid relators and hyperelliptic Lefschetz fibrations

We define a new invariant $w$ for hyperelliptic Lefschetz fibrations over closed oriented surfaces, which counts the number of Dirac braids included intrinsically in the monodromy, by using chart description introduced by the second author. As an application, we prove that two hyperelliptic Lefschetz fibrations of genus $g$ over a given base space are stably isomorphic if and only if they have the same numbers of singular fibers of each type and they have the same value of $w$ if $g$ is odd. We also give examples of pair of hyperelliptic Lefschetz fibrations with the same numbers of singular fibers of each type which are not stably isomorphic.Comment: 30 pages, 32 figures; (v2) the title changed, Section 1 rewritten, remarks added. arXiv admin note: text overlap with arXiv:1403.794

Virtual links which are equivalent as twisted links

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 non-orientable) 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