Narrow Results

Let $S=S(n)$ denote the infinite-type surface with $n$ ends, $n\in \mathbb{N}$, accumulated by genus. For $n\ge 6$, we show that the mapping class group of $S$ is topologically generated by five involutions. When $n\ge 3$, it is topologically generated by six involutions.

We say that a finite group G satisfies the independence property if, for every pair of distinct elements x and y of G, either $\{x,y\}$ is contained in a minimal generating set for G or one of x and y is a power of the other. We give a complete classification of the finite groups with this property, and in particular prove that every such group is supersoluble. A key ingredient of our proof is a theorem showing that all but three finite almost simple groups H contain an element s such that the maximal subgroups of H containing s, but not containing the socle of H, are pairwise non-conjugate.

We show that a certain natural class of tangles 'generate the collection of all tangles with respect to composition'. This result is motivated by, and describes the reasoning behind, the 'uniqueness assertion' in Jones' theorem on the equivalence between extremal subfactors of finite index and what we call 'subfactor planar algebras' here. This result is also used to identify the manner in which the planar algebras corresponding to M⊂M₁ and N^op⊂M^op are obtained from that of N⊂M.

Our results also show that 'duality' in the category of extremal subfactors of finite index extends naturally to the category of 'general' planar algebras (not necessarily finite-dimensional or spherical or connected or C*, in the terminology of Jones).

Polyak proved that the set $\{\mathrm{\Omega }1a,\mathrm{\Omega }1b,\mathrm{\Omega }2a,\mathrm{\Omega }3a\}$ is a minimal generating set of oriented Reidemeister moves. One may distinguish between forward and backward moves, obtaining $32$ different types of moves, which we call directed oriented Reidemeister moves. In this paper, we prove that the set of eight directed Polyak moves $\{\mathrm{\Omega }{1a}^{\uparrow },\mathrm{\Omega }{1a}^{\downarrow },\mathrm{\Omega }{1b}^{\uparrow },\mathrm{\Omega }{1b}^{\downarrow },\mathrm{\Omega }{2a}^{\uparrow },\mathrm{\Omega }{2a}^{\downarrow },\mathrm{\Omega }{3a}^{\uparrow },\mathrm{\Omega }{3a}^{\downarrow }\}$ is a minimal generating set of directed oriented Reidemeister moves. We also specialize the problem, introducing the notion of a $L$-generating set for a link $L$. The same set is proven to be a minimal $L$-generating set for any link $L$ with at least two components. Finally, we discuss knot diagram invariants arising in the study of $K$-generating sets for an arbitrary knot $K$, emphasizing the distinction between ascending and descending moves of type $\mathrm{\Omega }3$.

Let S be a generating set of a group G. We say that G has finite width relative to S if G = (S ∪ S^-1)^k for a suitable natural number k. We say that a group G is a group of finite C-width if G has finite width with respect to all conjugation-invariant generating sets of G. We give a number of examples of groups of finite C-width, and, in particular, we prove that the commutator subgroup F′ of Thompson's group F is a group of finite C-width. We also study the behavior of the class of all groups of finite C-width under some group-theoretic constructions; it is established, for instance, that this class is closed under formation of group extensions.

The symmetric group S_n and the alternating group A_n are groups of permutations on the set {0, 1, 2, …, n - 1} whose elements can be represented as products of disjoint cycles (the representation is unique up to the order of the cycles). In this paper, we show that whenever n ≥ k ≥ 2, the collection of all k-cycles generates S_n if k is even, and generates A_n if k is odd. Furthermore, we algorithmically construct generating sets for these groups of smallest possible size consisting exclusively of k-cycles, thereby strengthening results in [O. Ben-Shimol, The minimal number of cyclic generators of the symmetric and alternating groups, Commun. Algebra35(10) (2007) 3034–3037]. In so doing, our results find importance in the context of theoretical computer science, where efficient generating sets play an important role.

We provide three generalizations of all best known generating sets for $S_n$ and $A_n$: (1) $m$-cycles with consecutive entries; (2) $m$-cycles with fixed first $\ell$ entries; (3) two cycles, either $(1,\dots ,m),(1,\dots ,n)$ or $(1,\dots ,m),(2,\dots ,n)$. Our work on (3) fills a gap in the history of the problem, going back to a 1939 paper by S. Piccard [Sur les bases du groupe symétrique et du groupe alternant, Math. Ann. 116(1) (1939) 752–767].