Narrow Results

Using Yoshikawa's surface diagram, we constructed new invariants of ambient isotopy classes of smoothly embedded closed surfaces in ℝ⁴ via a state-sum model similar to the Kauffman's state-sum model for the Jones polynomial for classical knots and links in ℝ³. It is shown that the invariants can also be defined by skein relation and thus they are calculated from Yoshikawa's surface diagrams recurrently. Some of the properties of the invariants are given and explicit computations for several surfaces are included.

A. S. Lipson constructed two state models yielding the same classical link invariant obtained from the Kauffman polynomial F(a, u). In this paper, we apply Lipson's state models to marked graph diagrams of surface-links, and observe when they induce surface-link invariants.

A marked graph diagram is a link diagram possibly with marked 4-valent vertices. S. J. Lomonaco, Jr. and K. Yoshikawa introduced a method of representing surface-links by marked graph diagrams. Specially, K. Yoshikawa suggested local moves on marked graph diagrams, nowadays called Yoshikawa moves. It is now known that two marked graph diagrams representing equivalent surface-links are related by a finite sequence of these Yoshikawa moves. In this paper, we provide some generating sets of Yoshikawa moves on marked graph diagrams representing unoriented surface-links, and also oriented surface-links. We also discuss independence of certain Yoshikawa moves from the other moves.

In this article, we introduce a method to construct invariants of the stably equivalent surface links in ℝ⁴ by using invariants of classical knots and links in ℝ³. We give invariants derived from this construction with the Kauffman bracket polynomial.