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THE ADDITIVITY OF CROSSING NUMBERS

YUANAN DIAO

Department of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA

https://doi.org/10.1142/S0218216504003524Cited by:13 (Source: Crossref)

Abstract

It has long been conjectured that the crossing numbers of links are additive under the connected sum of links. This is a difficult problem in knot theory and has been open for more than 100 years. In fact, many questions of much weaker conditions are still open. For instance, it is not known whether Cr(K₁#K₂)≥Cr(K₁) or Cr(K₁#K₂)≥Cr(K₂) holds in general, here K₁#K₂ is the connected sum of K₁ and K₂ and Cr(K) stands for the crossing number of the link K. However, for alternating links K₁ and K₂, Cr(K₁#K₂)=Cr(K₁)+Cr(K₂) does hold. On the other hand, if K₁ is an alternating link and K₂ is any link, then we have Cr(K₁#K₂)≥Cr(K₁). In this paper, we show that there exists a wide class of links over which the crossing number is additive under the connected sum operation. This class is different from the class of all alternating links. It includes all torus knots and many alternating links. Furthermore, if K₁ is a connected sum of any given number of links from this class and K₂ is a non-trivial knot, we prove that Cr(K₁#K₂)≥Cr(K₁)+3.

Keywords:

AMSC: Primary 57M25

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