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.

We often study surface links in 4-space by using their projections into 3-space, or their broken surface diagrams. It is well-known that a broken surface diagram recovers the given surface link. In this paper, we study surface links in 4-space by using their generic projections into the plane. These projections have fold points and cusps as their singularities in general. We consider the question whether such a generic planar projection can recover the given surface link. We introduce the notion of banded braids, and show that a generic planar projection together with banded braids associated to the segments of the fold curve image can recover the given surface link. As an application, we give a new proof to the Whitney congruence concerning the normal Euler number of surface links.

We consider a surface link in the 4-space which can be presented by a simple branched covering over the standard torus, which we call a torus-covering link. Torus-covering links include spun T²-knots and turned spun T²-knots. In this paper we braid a torus-covering link over the standard 2-sphere. This gives an upper estimate of the braid index of a torus-covering link. In particular we show that the turned spun T²-knot of the torus (2, p)-knot has the braid index four.

This is the first step of the two steps to enumerate the minimal charts with two crossings. For a label $m$ of a chart $\mathrm{\Gamma }$ we denote by ${\mathrm{\Gamma }}_m$ the union of all the edges of label $m$ and their vertices. For a minimal chart $\mathrm{\Gamma }$ with exactly two crossings, we can show that the two crossings are contained in ${\mathrm{\Gamma }}_{\alpha }\cap {\mathrm{\Gamma }}_{\beta }$ for some labels $\alpha <\beta$. In this paper, we study the structure of a disk $D$ not containing any crossing but satisfying $\mathrm{\Gamma }\cap \partial D\subset {\mathrm{\Gamma }}_{\alpha +1}\cup {\mathrm{\Gamma }}_{\beta -1}$.

Let $\mathrm{\Gamma }$ be a chart, and we denote by ${\mathrm{\Gamma }}_m$ the union of all the edges of label $m$. A chart $\mathrm{\Gamma }$ is of type $(3,2,2)$ if there exists a label $m$ such that $w(\mathrm{\Gamma })=7$, $w({\mathrm{\Gamma }}_m\cap {\mathrm{\Gamma }}_{m+1})=3$, $w({\mathrm{\Gamma }}_{m+1}\cap {\mathrm{\Gamma }}_{m+2})=2$, and $w({\mathrm{\Gamma }}_{m+2}\cap {\mathrm{\Gamma }}_{m+3})=2$ where $w(G)$ is the number of white vertices in $G$. In this paper, we prove that there is no minimal chart of type $(3,2,2)$.

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let $\mathrm{\Gamma }$ be a chart, and we denote by ${\mathrm{\Gamma }}_m$ the union of all the edges of label $m$. A chart $\mathrm{\Gamma }$ is of type $(7)$ if there exists a label $m$ such that $w(\mathrm{\Gamma })=7$, $w({\mathrm{\Gamma }}_m\cap {\mathrm{\Gamma }}_{m+1})=7$ where $w(G)$ is the number of white vertices in $G$. In this paper, we prove that there is no minimal chart of type $(7)$.

The arrow polynomial is an invariant of framed oriented virtual links that generalizes the virtual Kauffman bracket. In this paper, we define the homological arrow polynomial, which generalizes the arrow polynomial to framed oriented virtual links with labeled components. The key observation is that, given a link in a thickened surface, the homology class of the link defines a functional on the surface’s skein module, and by applying it to the image of the link in the skein module this gives a virtual link invariant. We give a graphical calculus for the homological arrow polynomial by taking the usual diagrams for the Kauffman bracket and including labeled “whiskers” that record intersection numbers with each labeled component of the link. We use the homological arrow polynomial to study $(\mathbb{Z}/n\mathbb{Z})$-nullhomologous virtual links and checkerboard colorability, giving a new way to complete Imabeppu’s characterization of checkerboard colorability of virtual links with up to four crossings. We also prove a version of the Kauffman–Murasugi–Thistlethwaite theorem that the breadth of an evaluation of the homological arrow polynomial for an “h-reduced” diagram $D$ is $4(c(D)-g(D)+1)$.

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.