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AVERAGE LINKING NUMBERS GONZALO CONTRERAS, RENATO ITURRIAGA
 

Summary: AVERAGE LINKING NUMBERS
GONZALO CONTRERAS, RENATO ITURRIAGA
To the memory of Ricardo Ma~n'e.
1. Introduction.
Let 1 fl 1 ; fl 2 be two disjoint oriented knots in the three dimensional
sphere S 3 the linking number `(fl 1 ; fl 2 ) is defined, as the algebraic in­
tersection number of fl 1 with an orientable surface N transversal to fl 1 ,
with oriented boundary fl 2 . For better understanding of our results we
need the two following interpretations of the linking number (See [R]):
1) For any regular planar projection it is the algebraic crossing num­
ber of fl 1 over fl 2 .
2) It is the degree of the map
T : T 2 ! S 2
(t 1 ; t 2 ) ! fl 1 (t 1 ) \Gamma fl 2 (t 2 )
jfl 1 (t 1 ) \Gamma fl 2 (t 2 )j
where T 2 is the two dimensional torus and S 2 is the two dimensional
sphere. Hence
`(fl 1 ; fl 2 ) = 1
vol(S 2 )
Z

  

Source: Acevedo, Renato Iturriaga - Centro de Investigación en Matemáticas (CIMAT)

 

Collections: Mathematics