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Invariants of knots, surfaces in R3, and foliations

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Abstract

We give a survey of some known results related to combinatorial and geometric properties of finite-order invariants of knots in a three-dimensional space. We study the relationship between Vassiliev invariants and some classical numerical invariants of knots and point out the role of surfaces in the investigation of these invariants. We also consider combinatorial and geometric properties of essential tori in standard position in closed braid complements by using the braid foliation technique developed by Birman, Menasco, and other authors. We study the reductions of link diagrams in the context of finding the braid index of links.

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Authors and Affiliations

  1. Institute for Applied Problems in Mechanics and Mathematics, Ukrainian National Academy of Sciences, Lviv

    L. P. Plakhta

  2. Institute of Mathematics, Gdańsk University, Poland

    L. P. Plakhta

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  1. L. P. Plakhta
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 9, pp. 1239–1252, September, 2007.

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Plakhta, L.P. Invariants of knots, surfaces in R3, and foliations. Ukr Math J 59, 1385–1396 (2007). https://doi.org/10.1007/s11253-007-0093-1

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  • DOI: https://doi.org/10.1007/s11253-007-0093-1

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