71 research outputs found
The Alexander polynomial of (1,1)-knots
In this paper we investigate the Alexander polynomial of (1,1)-knots, which
are knots lying in a 3-manifold with genus one at most, admitting a particular
decomposition. More precisely, we study the connections between the Alexander
polynomial and a polynomial associated to a cyclic presentation of the
fundamental group of an n-fold strongly-cyclic covering branched over the knot,
which we call the n-cyclic polynomial. In this way, we generalize to all
(1,1)-knots, with the only exception of those lying in S^2\times S^1, a result
obtained by J. Minkus for 2-bridge knots and extended by the author and M.
Mulazzani to the case of (1,1)-knots in the 3-sphere. As corollaries some
properties of the Alexander polynomial of knots in the 3-sphere are extended to
the case of (1,1)-knots in lens spaces.Comment: 11 pages, 1 figure. A corollary has been extended, and a new example
added. Accepted for publication on J. Knot Theory Ramification
All strongly-cyclic branched coverings of (1,1)-knots are Dunwoody manifolds
We show that every strongly-cyclic branched covering of a (1,1)-knot is a
Dunwoody manifold. This result, together with the converse statement previously
obtained by Grasselli and Mulazzani, proves that the class of Dunwoody
manifolds coincides with the class of strongly-cyclic branched coverings of
(1,1)-knots. As a consequence, we obtain a parametrization of (1,1)-knots by
4-tuples of integers. Moreover, using a representation of (1,1)-knots by the
mapping class group of the twice punctured torus, we provide an algorithm which
gives the parametrization of all torus knots.Comment: 22 pages, 19 figures. Revised version with minor changes in
Proposition 5. Accepted for publication in the Journal of the London
Mathematical Societ
(1,1)-knots via the mapping class group of the twice punctured torus
We develop an algebraic representation for (1,1)-knots using the mapping
class group of the twice punctured torus MCG(T,2). We prove that every
(1,1)-knot in a lens space L(p,q) can be represented by the composition of an
element of a certain rank two free subgroup of MCG(T,2) with a standard element
only depending on the ambient space. As a notable examples, we obtain a
representation of this type for all torus knots and for all two-bridge knots.
Moreover, we give explicit cyclic presentations for the fundamental groups of
the cyclic branched coverings of torus knots of type (k,ck+2).Comment: 18 pages, 10 figures. New version with minor changes. Accepted for
publication in Advances in Geometr
Representations of (1,1)-knots
We present two different representations of (1,1)-knots and study some
connections between them. The first representation is algebraic: every
(1,1)-knot is represented by an element of the pure mapping class group of the
twice punctured torus. The second representation is parametric: every
(1,1)-knot can be represented by a 4-tuple of integer parameters. The strict
connection of this representation with the class of Dunwoody manifolds is
illustrated. The above representations are explicitly obtained in some
interesting cases, including two-bridge knots and torus knots.Comment: 16 pages, 7 figures. To appear in Fundamenta Mathematicae, special
volume Proceedings of Knots in Poland, vol. I
A Markov theorem for generalized plat decomposition
We prove a Markov theorem for tame links in a connected closed orientable
3-manifold with respect to a plat-like representation. More precisely,
given a genus Heegaard surface for we represent each link in
as the plat closure of a braid in the surface braid group
and analyze how to translate the equivalence
of links in under ambient isotopy into an algebraic equivalence in
. First, we study the equivalence problem in ,
and then, to obtain the equivalence in , we investigate how isotopies
corresponding to "sliding" along meridian discs change the braid
representative. At the end we provide explicit constructions for Heegaard genus
1 manifolds, i.e. lens spaces and .Comment: Acknowledgements added. Accepted for publication on Ann. Sc. Norm.
Super. Pisa Cl. Sc
Extending homeomorphisms from punctured surfaces to handlebodies
Let be a genus handlebody and
be the group of the isotopy classes of
orientation preserving homeomorphisms of ,
fixing a given set of points. In this paper we find a finite set of
generators for , the subgroup of
consisting of the isotopy classes of
homeomorphisms of admitting an extension to the handlebody and
keeping fixed the union of disjoint properly embedded trivial arcs. This
result generalizes a previous one obtained by the authors for . The
subgroup turns out to be important for the study of knots
and links in closed 3-manifolds via -decompositions. In fact, the links
represented by the isotopy classes belonging to the same left cosets of
in are equivalent.Comment: We correct the statements of Theorem 9 and 10, by adding missing
generators, and improve the statement of Theorem 10, by removing some
redundant generator
Knot quandle decompositions
We show that the fundamental quandle defines a functor from the oriented
tangle category to a suitably defined quandle category. Given a tangle
decomposition of a link , the fundamental quandle of may be obtained
from the fundamental quandles of tangles. We apply this result to derive a
presentation of the fundamental quandle of periodic links, composite knots and
satellite knots.Comment: 23 pages, 12 figure
Extending homeomorphisms from 2-punctured surfaces to handlebodies
Let F a closed connected orientable surface bounding a genus g handlebody H.
In this paper we find a finite set of generators for the subgroup E(2,g) of the
pure mapping class group of the twice punctured torus PMCG(2,g), consisting of
the isotopy classes of homeomorphisms of F which admit an extension to H
keeping a properly embedded trivial arc fixed. This subgroup turns out to be
important for the study of knots in closed 3-manifolds via (g,1)-decomposition.
In fact, the knots represented by the isotopy classes belonging to the same
left cosets of E(2,g) in PMCG(2,g) are equivalent.Comment: 9 pages, 5 figure
Strongly-cyclic branched coverings of (1,1)-knots and cyclic presentations of groups
We study the connections among the mapping class group of the twice punctured
torus, the cyclic branched coverings of (1,1)-knots and the cyclic
presentations of groups. We give the necessary and sufficient conditions for
the existence and uniqueness of the n-fold strongly-cyclic branched coverings
of (1,1)-knots, through the elements of the mapping class group. We prove that
every n-fold strongly-cyclic branched covering of a (1,1)-knot admits a cyclic
presentation for the fundamental group, arising from a Heegaard splitting of
genus n. Moreover, we give an algorithm to produce the cyclic presentation and
illustrate it in the case of cyclic branched coverings of torus knots of type
(k,hk+1) and (k,hk-1).Comment: 16 pages, 2 figures. to appear in the Mathematical Proceedings of the
Cambridge Philosophical Societ
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