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Crossing changes Martin Scharlemann1
 

Summary: Crossing changes
Martin Scharlemann1
A naive but often useful way of thinking of a knot or link in R3 is to generically project
it onto a plane, keeping account of which part of the knot goes under and which part goes
over at any given crossing. Think of laying the knot on a table and taking note of how it
crosses itself whenever one part of it lies on top of another. It's natural to ask how the
knot is changed by altering one of the crossings, reversing which arc goes over and which
goes under at one of the crossing points.
This survey article is meant to explore this question. Though it's a naive question, it
connects at a deep level to some of the most important ideas in modern low-dimensional
topology.
1 Unknotting number, tunnel number, crossing number
Begin by making a distinction: Given a knot, one can consider what happens when one
changes a crossing in a particular projection; or, more generally, one can ask what happens
when one changes a crossing in some unspecified projection of the knot. The distinction is
important, especially when special types of projections are being considered. For example,
an alternating knot is a knot which has a projection (called an alternating projection) in
which over- and under- crossings alternate as one travels a circuit around the knot. A
crossing change of an alternating knot may or may not be one which can be realized in an
alternating projection. A minimal projection of a knot is one which minimizes the number

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics