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An invariant of regular isotopy. (English) Zbl 0763.57004
Two links in the 3-sphere are said to be regular isotopic if one is transformed into the other by a finite sequence of Reidemeister moves of type II and III (excluding of type I). One of the objectives of this paper is to demonstrate that polynomial invariants defined by skein relations are, in fact, obtained as certain normalization of regular isotopy invariants. The Jones polynomial, Kauffman polynomial and Dubrovnik polynomial are examples of this type . In the later section, the author gives a complete proof of the well-definedness and uniqueness of the L-polynomial as a regular isotopy invariant. Another objective is to give ingenious diagrammatic interpretation of several algebras, e.g. a braid monoid, connection monoid, Brauer monoid. These monoids and their interpretations have been used extensively in recent knot theory. This paper is rather a research oriented expository paper than a pure research paper.

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M15 Relations of low-dimensional topology with graph theory
82B23 Exactly solvable models; Bethe ansatz
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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