v 1 2 4 A pr 1 99 5 A Link Invariant from Quantum Dilogarithm
The link invariant, arising from the cyclic quantum dilogarithm via the particular R-matrix construction, is proved to coincide with the invariant of triangulated links in S introduced in [14]. The obtained invariant, like Alexander-Conway polynomial, vanishes on disjoint union of links. The R-matrix can be considered as the cyclic analog of the universal R-matrix associated with Uq(sl(2)) algebra.
An enumeration of knots and links, and some of their algebraic properties
Publisher Summary This chapter describes knots and links, and some of their algebraic properties. An edge-connected 4-valent planar map is called a polyhedron, and a polyhedron is basic if no region has just 2 vertices. The term region includes the infinite region, which is regarded in the same light as the others. Knot diagrams can be obtained from polyhedra by substituting tangles for their vertices, for instance, tangles 1 or −1 could always be substituted. A knot diagram K can be obtained by substituting algebraic tangles for the vertices of some nonbasic polyhedron P. There is a polyhedron Q with fewer vertices than P obtained by shrinking some 2-vertex region of P, and K can simply be obtained by substituting algebraic tangles for the vertices of Q. Any knot diagram can be obtained by substituting algebraic tangles for the vertices of some basic polyhedron P in fact P, and the manner of substitution is essentially unique.
Dilogarithm identities, q-difference equations and the Virasoro algebra
We give a new proof of the dilogarithm identities, associated to the (2,2n+1) minimal models of the Virasoro algebra.