Simplified numerical form of universal finite type invariant of Gauss words

In the present paper, we study the finite type invariants of Gauss words. In the Polyak algebra techniques, we reduce the determination of the group structure to transformation of a matrix into its Smith normal form and we give the simplified form of a universal finite type invariant by means of the isomorphism of this transformation. The advantage of this process is that we can implement it as a computer program. We obtain the universal finite type invariant of degree 4, 5, and 6 explicitly. Moreover, as an application, we give the complete classification of Gauss words of rank 4 and the partial classification of Gauss words of rank 5 where the distinction of only one pair remains.Comment: 12 pages, 3 table

Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory

This paper is a concise introduction to virtual knot theory, coupled with a list of research problems in this field.Comment: 65 pages, 24 figures. arXiv admin note: text overlap with arXiv:math/040542