834 research outputs found
A new algorithm for recognizing the unknot
The topological underpinnings are presented for a new algorithm which answers
the question: `Is a given knot the unknot?' The algorithm uses the braid
foliation technology of Bennequin and of Birman and Menasco. The approach is to
consider the knot as a closed braid, and to use the fact that a knot is
unknotted if and only if it is the boundary of a disc with a combinatorial
foliation. The main problems which are solved in this paper are: how to
systematically enumerate combinatorial braid foliations of a disc; how to
verify whether a combinatorial foliation can be realized by an embedded disc;
how to find a word in the the braid group whose conjugacy class represents the
boundary of the embedded disc; how to check whether the given knot is isotopic
to one of the enumerated examples; and finally, how to know when we can stop
checking and be sure that our example is not the unknot.Comment: 46 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTVol2/paper9.abs.htm
Multifraction reduction III: The case of interval monoids
We investigate gcd-monoids, which are cancellative monoids in which any two
elements admit a left and a right gcd, and the associated reduction of
multifractions (arXiv:1606.08991 and 1606.08995), a general approach to the
word problem for the enveloping group. Here we consider the particular case of
interval monoids associated with finite posets. In this way, we construct
gcd-monoids, in which reduction of multifractions has prescribed properties not
yet known to be compatible: semi-convergence of reduction without convergence,
semi-convergence up to some level but not beyond, non-embeddability into the
enveloping group (a strong negation of semi-convergence).Comment: 23 pages ; v2 : cross-references updated ; v3 : one example added,
typos corrected; final version due to appear in Journal of Combinatorial
Algebr
- …