Towards invariants of surfaces in $4$-space via classical link invariants

Abstract

Abstract. In this paper, we introduce a method to construct ambient isotopy invariants for smooth imbeddings of closed surfaces into 4-space by using hyperbolic splittings of the imbedded surfaces and an arbitrary given isotopy or regular isotopy invariant of classical knots and links in 3-space. Using this construction, adopting the Kauffman bracket polynomial as an example, we produce some invariants.

Bikei homology

“…Such a diagram represents a knots closed surface if both resolutions of the saddle yield unlinks; otherwise, the diagram represents a cobordism between the links represented by the smoothed diagrams. Two such diagrams represent ambient isotopic knotted surfaces if and only if they are related by a sequence of the Reidemeister moves together with the Yoshikawa moves See for instance [5,14,15,16,17] for more.…”

Polynomial of an oriented surface-link diagram via quantum A2 invariant

“…In [15], the fourth author introduced a method of constructing invariant for a surface-link by means of a marked graph diagram and a state-sum model associated to a classical link invariant as its state evaluation. In this paper, we define a polynomial in Z[a −1 , a, x, y] for an oriented marked graph diagram by using the A 2 bracket · A 2 in the line of [15] and study how the polynomial changes under Yoshikawa moves. In the process of this argument, the notion of a ribbon marked graph is introduced to show that this polynomial is useful for an invariant of a ribbon 2-knot.…”

Computations of quandle cocycle invariants of surface-links using marked graph diagrams

“…So one can use marked graph diagrams for studying surface-links and their invariants (cf. [1,13,14,22,23,24,25,26,30]).…”