Narrow Results

Carter, Jelsovsky, Kamada, Langford and Saito introduced the quandle cocycle invariants of 2-knots, and calculated the cocycle invariant of a 2-twist-spun trefoil knot associated with a 3-cocycle of the dihedral quandle of order 3. Asami and Satoh calculated the cocycle invariants of twist-spun torus knots τ^rT(m,n) associated with 3-cocycles of some dihedral quandles. They used tangle diagrams of the torus knots. In this paper, we calculate the cocycle invariants of twist-spun 2-bridge knots τ^rS(α,β) by a similar method.

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.

We consider a surface embedded in the 3-space. Such a surface is called unknotted if the closures of the complementary regions in the 3-sphere are both handlebodies. In this paper, we give a sufficient condition for a torus in the 3-space to be unknotted by using a projection onto a plane. We also prove that this condition is not sufficient for surfaces of higher genera.

In [Towards invariants of surfaces in 4-space via classical link invariants, Trans. Amer. Math. Soc.361 (2009) 237–265], Lee defined a polynomial [[D]] for marked graph diagrams D of surface-links in 4-space by using a state-sum model involving a given classical link invariant. In this paper, we deal with some obstructions to obtain an invariant for surface-links represented by marked graph diagrams D by using the polynomial [[D]] and introduce an ideal coset invariant for surface-links, which is defined to be the coset of the polynomial [[D]] in a quotient ring of a certain polynomial ring modulo some ideal and represented by a unique normal form, i.e. a unique representative for the coset of [[D]] that can be calculated from [[D]] with the help of a Gröbner basis package on computer.

Carrell defined the fundamental biquandle of an oriented surface-link by a presentation obtained from its broken surface diagram, which is an invariant up to isomorphism of the fundamental biquandle. Ashihara gave a method to calculate the fundamental biquandle of an oriented surface-link from its marked graph diagram (ch-diagram). In this paper, we discuss the fundamental Alexander biquandles of oriented surface-links via marked graph diagrams, derived computable invariants and their applications to detect non-invertible oriented surface-links.

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.

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.