Narrow Results

The main result of this paper is a proof of existence of a nontrivial knot on any embedded template, that was left as an open question to prove in [Ghrist et al., 1997] without using the Bennequin's inequality [Ghrist et al., 1997]. This result in the branched two-manifold case, which we prove by a sequence of lemmas showing our simple template (or ones with twists) containing nontrivial knots is (are) contained in every template as a subtemplate, enables us to generalize it later in this paper to certain forms of three-templates in four-dimensional dynamical systems by simply using the technique of "spinning" the knots in the lower dimensional templates to obtain the spun knotted surfaces.

In this paper we consider spun dynamical systems and show that we can obtain a system on S³ × S¹ which contains any finite set of knotted and linked surfaces which are invariant surfaces for the flow.

We define a knot invariant and a 2-knot invariant from any finite categorical group. We calculate an explicit example for the Spun Trefoil.

The singularity set of a generic standard projection to the three space of a closed surface linked in four space, consists of at most three types: double points, triple points or branch points. We say that this generic projection image is p-diagram if it does not contain any triple point. Two p-diagrams of equivalent surface links are called p-equivalent if there exist a finite sequence of local moves, such that each of them is one of the four moves taken from the seven on the well known Roseman list, that connects only p-diagrams. It is natural to ask that whether any of two p-diagrams of equivalent surface links always p-equivalent? We introduce an invariant of p-equivalent diagrams and an example of linked surfaces that answers our question negatively.

We will discuss a method for visual presentation of knotted surfaces in the four space, by examining a number and a position of its Morse’s critical points. Using this method, we will investigate surface-knot with one critical point of index $1$. Then we show infinitely many mutually distinct surface-knots that have an embedding with two critical points of index $1$. Next we define a long flat form of a banded link for any surface-knot and show diagrammatically a long flat form of $n$-twist-spun $(2,t)$-torus knots.

Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in ${ℝ}^4$; including virtual crossings allows for a theory of virtual knotted surfaces and virtual cobordisms. Biquandle counting invariants are defined only for marked vertex diagrams representing knotted orientable surfaces; we extend these invariants to all virtual marked vertex diagrams by considering colorings by involutory biquandles, also known as bikei. We introduce a way of representing marked vertex diagrams with Gauss diagrams and use these to characterize orientability.

We present a marked analog of Carter and Saito’s movie theorem. Our definition of marking was chosen to coincide with the markings that arise in link Floer homology. In order to deal with complications arising from certain isotopes, we define three equivalences for marked surfaces and work over an equivalence class of marked surfaces when proving our generalization of Carter and Saito’s movie theorem.

In this paper, we consider local moves on classical and welded diagrams of string links, and the notion of welded extension of a classical move. Such extensions being non-unique in general, the idea is to find a topological criterion which could isolate one extension from the others. To that end, we turn to the relation between welded string links and knotted surfaces in ${ℝ}^4$, and the ribbon subclass of these surfaces. This provides a topological interpretation of classical local moves as surgeries on surfaces, and of virtual local moves as surgeries on ribbon surfaces. Comparing these surgeries leads to the notion of ribbon residue of a classical local move, and we show that up to some broad conditions there can be at most one welded extension which is a ribbon residue. We provide three examples of ribbon residues, for the self-crossing change, the Delta and the band-pass moves. However, for the latter, we note that the given residue is actually not an extension of the band-pass move, showing that a classical move may have a ribbon residue and a welded extension, but no ribbon residue which is an extension.

A spun knotted torus in the 4-sphere is formed by rigidly sweeping a knotted curve along a circle. Alternately, as the knotted curve is swept along the circle we could give it a number of full turns (Dehn twists). We show the resulting knotted torus depends only on the knotted curve and whether the number of turns is even or odd. The even and odd turned spun tori have nondiffeomorphic complements. This is generalized in some cases to include twist spun turned torus knots.

Let F be an embedded Klein bottle in S⁴\{∞}. If the singular set Γ(F*) of the projection F* of F into R³ consists of at most three disjoint simple dosed curves, then F bounds a said Klein bottle in the 4-sphere S⁴, i.e., F can be moved to the standard Klein bottle.

A survey of our results in the diagrammatic approaches to knotted surfaces is given in part I of this paper. Results obtained by knotted surface diagrams and by movie descriptions will be reviewed. Diagrammatic ways of solving generalizations of the Yang-Baxter equation are explained. In part II, we give smoothings of higher dimensional knots that topologically generalize the classical knot smoothings of crossings and Kauffman's bracket trick for resolving the Yang-Baxter equation.