119 research outputs found
Corrigendum to “Extending homeomorphisms from punctured surfaces to handlebodies” [Topology Appl. 155 (6) (2008) 610–621]
No full textAbstractWe correct the statements of Theorems 9 and 10 of [A. Cattabriga, M. Mulazzani, Extending homeomorphisms from punctured surfaces to handlebodies, Topology Appl. 155 (2008) 610–621], by adding missing generators, and improve the statement of Theorem 10, by removing some redundant generators
On the axioms of singquandles
No full textIn this paper, we deal with the notion of singquandles introduced in [I. R. U. Churchill, M. Elhamdadi, M. Hajij and S. Nelson, Singular knots and involutive quandles, J. Knot Theory Ramifications 26(14) (2017) 1750099]. This is an algebraic structure that naturally axiomatizes Reidemeister moves for singular links, similarly to what happens for ordinary links and quandles. We present a new axiomatization that shows different algebraic aspects and simplifies applications. We also reformulate and simplify the axioms for affine singquandles (in particular in the idempotent case)
The Alexander polynomial of (1,1)-knots
No full textIn 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 K, which we call the n-cyclic polynomial of K. In
this way, we generalize to all (1, 1)-knots, with the only exception of those lying in
S^2 × S^1, a result obtained by Minkus for 2-bridge knots and extended by the author
and M. Mulazzani to the case of (1, 1)-knots in S^3. As corollaries some properties of
the Alexander polynomial of knots in S^3 are extended to the case of (1, 1)-knots in lens
spaces
A Blended Teaching Sequence for Introducing Eigentheory in a Large University Class
No full textWe present the design and implementation of a teaching sequence on eigentheory that aims at encouraging at facilitating students’ active participation in a class attended by a high number of students. We describe how the use of the online noticing board Padlet facilitates the implementation of activities in small groups during lectures. These activities are designed based on the results of a pilot study aimed at assessing students’ understanding of eigentheory notions. We analyze various types of data collected during and after the implementation, showing a positive response from both the course teacher and the students to this teaching and learning methodology
Strongly-cyclic branched coverings of (1, 1)-knots and cyclic presentations of groups
No full textWe 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)
The complexity of orientable graph manifolds
No full textWe give an upper bound for the Matveev complexity of the whole class of closed connected orientable prime graph manifolds; this bound is sharp for all 14502 graph manifolds of the Recogniser catalogue (available at http://matlas.math.csu.ru/?page=search
Criteria for identifying knapping skill level through the analysis of lithic cores : An example from Val Lastari, Late Paleolithic, Italy
No full textThe application of knapping skill analysis has increased over the years, starting from the Eighties and, by now, being employed all over the world on different material cultures and chronological periods. We studied the cores reduced at Val Lastari, a Recent Epigravettian lithic workshop in north-eastern Italy, for recognizing the most influential technological variables to define stoneknapping behaviours. In our case, knapping accidents and cores appearance had more weight than criteria such as core preparation and raw material features thanks to the strategies of exploitation chosen by hunter-gatherers of Val Lastari, and to the ecological context they interacted with. This is the first time that a study of its kind has been proposed for an Italian site, becoming a unique possibility to enrich our knowledge about Epigravettian social learning and organization with new and stimulating information
Representations of (1,1)-knots
No full textWe 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 (a,b,c,r) 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
A Markov theorem for generalized plat decomposition
No full textWe 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
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