Simplifying branched covering surface-knots by an addition of 1-handles with chart loops

A branched covering surface-knot over an oriented surface-knot $F$ is a surface-knot in the form of a branched covering over $F$. A branched covering surface-knot over $F$ is presented by a graph called a chart on a surface diagram of $F$. For a branched covering surface-knot, an addition of 1-handles equipped with chart loops is a simplifying operation which deforms the chart to the form of the union of free edges and 1-handles with chart loops. We investigate properties of such simplifications.Comment: 26 pages, 15 figures, title changed, minor modifications, to appear in J. Knot Theory Ramification

Surface links with free abelian link groups

It is known that if a classical link group is a free abelian group, then its rank is at most two. It is also known that a $k$-component 2-link group ($k>1$) is not free abelian. In this paper, we give examples of $T^2$-links each of whose link groups is a free abelian group of rank three or four. Concerning the $T^2$-links of rank three, we determine the triple point numbers and we see that their link types are infinitely many.Comment: 10 pages, 6 figures, minor modifications, to appear in J. Math. Soc. Japa

Knotted surfaces constructed using generators of the BMW algebras and their graphical description

We consider knotted surfaces and their graphical description that include 2-dimensional braids and their chart description. We introduce a new construction of surfaces in $D^2 \times I \times I$, called BMW surfaces, that are described as the trace of deformations of tangles generated by generators $g_i, g_i^{-1}, e_i$ of the BMW (Birman-Murakami-Wenzl) algebras, where $g_i$ and $g_i^{-1}$ are a standard generator of the $n$-braid group and its inverse, and $e_i$ is a tangle consisting of a pair of "hooks". And we introduce a notion of BMW charts that are graphs in $I \times I$ and show that a BMW surface has a BMW chart description.Comment: 29 pages, 21 figure