115 research outputs found
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
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
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
Counting Dirac braid relators and hyperelliptic Lefschetz fibrations
We define a new invariant 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 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 if 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
Colorings and doubled colorings of virtual doodles
A virtual doodle is an equivalence class of virtual diagrams under an
equivalence relation generated by flat version of classical Reidemesiter moves
and virtual Reidemsiter 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
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