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Link polynomials and a graphical calculus. (English) Zbl 0795.57001
A 4-valent graph with rigid vertices (RV) can be regarded as an embedding of a graph whose vertices have been replaced by rigid disks. RV-isotopy (an affine motion of the disks with topological ambient isotopies of the strands) therefore preserves the cyclic ordering of the strand at each vertex. There is a set of Reidemeister-like moves which generate RV- isotopies. To each isotopy class of graphs is associated a collection of ambient isotopy classes of knots and links, invariant under rigid isotopy, constructed by replacing each disk with the four possible ways of connecting the strands at that disk. In this paper polynomial invariants of graphs are constructed from known invariants for knots and links. A general procedure is first given using any polynomial or scalar invariant of knots and kinks, assuming it behaves multiplicatively under Type I moves. The construction is then used with the Homfly and Kauffman polynomials. The Homfly (Laurent) polynomial is a two variable invariant of links. It is generalized to a 3-variable invariant of RV ambient isotopies for 4-valent graphs in three space. A calculus on planar graphs is given to compute recursively these polynomials. For braids there is a relationship between this graphical calculus and the generators of the Hecke algebra for the \(n\)-strand braid group. A similar development gives a generalization of the Dubrovnik polynomial, a variant of the Kauffman polynomial. In this case the graphical identities are related to the Birman-Wenzel algebra.

MSC:
57M15 Relations of low-dimensional topology with graph theory
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