Kako težko je razvozlati nevozel?

V tem članku je obravnavan eden od elementarnejših problemov iz teorije vozlov: če vozelni diagram predstavlja nevozel, kako težko je to pokazati z osnovnimi potezami vozlov — Reidemeistrovimi pomiki? Podani sta dve različni interpretaciji tega vprašanja: koliko „zapleten“ postane diagram med razvozlavanjem in koliko Reidemeistrovih pomikov je potrebnih za razvozlavanje. Poleg nekaterih osnovnih definicij teorije vozlov sta podani definiciji Morsejeve oblike in obločne predstavitve, ki omogočata kombinatorično obravnavo problema. Z uporabo izrekov Dynnikova in kombinatoričnih metod se izpelje zgornja meja za obe vprašanji.

How difficult is it to unknot an unknot?

This paper discusses one of the more elementary problems from knot theory: if a knot diagram represents the unknot, how difficult is it to show that via the elementary moves of knots — Reidemeister moves? Two different interpretations of this question are given: how “complicated” does a diagram become while unknotting, and how many Reidemeister moves are needed to unknot it? Along with some basic definitions of knot theory, the definitions of Morse form and arc-presentations are given, which allow for a combinatorial handling of the question. Using theorems of Dynnikov and combinatorial methods, upper bounds are derived for both questions.