Burnside obstructions to the Montesinos–Nakanishi 3–move conjecture

Burnside obstructions to the Montesinos – Nakanishi 3 – move conjecture

Yasutaka Nakanishi asked in 1981 whether a 3–move is an unknotting operation. In Kirby’s problem list, this question is called The Montesinos–Nakanishi 3– move conjecture. We define the nth Burnside group of a link and use the 3rd Burnside group to answer Nakanishi’s question; ie, we show that some links cannot be reduced to trivial links by 3–moves. AMS Classification numbers Primary: 57M27 Se...

Burnside Orders, Burnside Algebras and Partition Lattices

The permutation representation theory of groups has been extended, through quasigroups, to one-sided left (or right) quasigroups. The current paper establishes a link with the theory of ordered sets, introducing the concept of a Burnside order that generalizes the poset of conjugacy classes of subgroups of a finite group. Use of the Burnside order leads to a simplification in the proof of key p...

The Restricted Burnside Problem

In 1902 William Burnside [5] wrote 'A still undecided point in the theory of discontinuous groups is whether the order of a group may be not finite, while the order of every operation it contains is finite'. In modern terminology the most general form of the problem is 'can a finitely generated group be infinite while every element in the group has finite order?'. This question was answered in ...

Obstructions to Shiftedness

In this paper we show results on the combinatorial properties of shifted simplicial complexes. We prove two intrinsic characterization theorems for this class. The first theorem is in terms of a generalized vicinal preorder. It is shown that a complex is shifted if and only if the preorder is total. Building on this we characterize obstructions to shiftedness and prove there are finitely many i...

Obstructions to Homotopy Invariance