Posted on Categories:Low Dimensional Topology, 低维拓扑, 数学代写

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## 数学代写|低维拓扑代写Low Dimensional Topology代考|Moves that Change the Topology of the Underlying Foam

It is important to remark that not only are embedded trivalent graphs studied for their own sake, but a given knotted handlebody in 3-space deformation retracts to an embedded trivalent graph. The graph, however, is not unique. Two graphs that “carry” such a knotted handlebody differ by the so-called IH-move. Up to equivalence, the $\mathrm{IH}$-move is given via the movie parametrization of the basic foam $Y^2$. The theory of knotted handlebodies embedded in 3 -space is equivalent to the theory of knotted trivalent graphs modulo the IH-move.

A similar situation holds in 4-space. We can include among the Roseman moves two additional moves that are indicated below. The first of these is the invertibility of the IH-move. In the theory of special spines for 3-manifolds it is sometimes called the lune move or the orthogonality condition. The reason for the latter name comes from the Tureav-Viro [15] invariants, the neighborhood of a vertex of a foam is colored by representations of $U_q\left(s l_2\right)$, and the move corresponds to the orthogonality condition for the $6 j$-symbol. See also [4]. Observe that if a foam is embedded in 3-space, then regular neighborhood of the foam is invariant under this move. So similarly, a regular neighborhood in 4-space of such a foam is also invariant since it can be obtained from the neighborhood in 3-space by the cartesian product with an open interval.

## 数学代写|低维拓扑代写Low Dimensional Topology代考|Future Work

This paper is a technical piece that is necessary for a serious study of knotted foams and their 3 (and higher)-dimensional generalizations. In work with Atsushi Ishii and Masahico Saito, we will establish a cohomology theory for certain algebraic systems that is sufficient to define nontrivial invariants of knotted 2-foams in 4-space.

The inclusion of the penultimate section is also meant to indicate the initial stages in the study of 3-dimensional foams. In particular, one can construct movies of 3-dimensional foams embedded in 5 -space by including the Roseman/Reidemeister moves of Theorem 1.1, the orthogonality and Eilliott-Beidenharn moves, critical points of surfaces and critical points of the edge sets. Thus the critical points of surfaces correspond to $0,1,2$, and 3-handles that are attached to the solid sheets of 3 -foams. Moreover, the edge set of a 3 -foam is a 2-foam. Important moves to 3 -foams are easy to establish. A full Roseman-type theorem is unknown to the author at this time tedious.

Finally, it is worth mentioning that there is an underlying categorical motivation here that is related to the tangle hypothesis of Baez and Dolan [2]. Here we are considering the interaction between a braiding and a Frobenius structure as well as the identities among relations of these. The precise location of knotted foams in the BaezDolan table is an interesting taxonomic problem.

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