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TRANSACTIONSOF THE AMERICAN MATHEMATICAL SOCIETY
 

Summary: TRANSACTIONSOF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 336, Number 2, April 1993
UNLINKING VIA SIMULTANEOUSCROSSING CHANGES
MARTIN SCHARLEMANN
ABSTRACT.Given two distinct crossings of a knot or link projection, we consider
the question: Under what conditions can we obtain the unlink by changing both
crossings simultaneously? More generally, for which simultaneous twistings at
the crossings is the genus reduced? Though several examples show that the
answer must be complicated, they also suggest the correct necessary conditions
on the twisting numbers.
Let L be an oriented link in s3with a generic projection onto the plane R~.
Let a be a short arc in R2 transverse to both strands of L at a crossing, so that
the strands pass through a in opposite directions. Then the inverse image of
a contains a disk punctured twice, with opposite orientation, by L. Define a
crossing disk D for a link L in S3 to be a disk which intersects L in precisely
two points, of opposite orientation. It is easy to see that any crossing disk arises
in the manner described. Twisting the link q times as it passes through D is
equivalent to doing l l q surgery on dD in S3 and adds 29 crossings to this
projection of L . We say that this new link L(q) is obtained by adding q twists

  

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

 

Collections: Mathematics