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Independence of Roseman moves including triple points. (English) Zbl 1350.57028
A surface-link is a submanifold of Euclidean 4-space \(\mathbb{R}^4\) diffeomorphic to a closed surface. Two surface-links are equivalent if one can be deformed into the other by an isotopy of \(\mathbb{R}^4\). Two surface-links are equivalent if and only if their diagrams are related by ambient isotopies of \(\mathbb{R}^3\) and a finite sequence of seven types of local modifications called Roseman moves [D. Roseman, Banach Cent. Publ. 42, 347–380 (1998; Zbl 0906.57010)]. Among the seven types of Roseman moves, three types involve triple points; in the paper, the authors call them the moves of type T1, T2 and BT. The move of type T2 is also called the tetrahedral move. For a subset \(S\) of the set of the types of Roseman moves, two diagrams of a surface-link are said to be \(S\)-dependent if any sequence of Roseman moves relating them contains at least one move in \(S\).
The main results are as follows. For each diagram \(D\) of any surface-link \(F\), there is another diagram \(D'\) of \(F\) which has the same number of triple points as \(D\), but \(D\) and \(D'\) are \(\{\)T1, T2\(\}\)-dependent. There is an \(S^2\)-knot with a pair of \(\{\)T2\(\}\)-dependent diagrams.
The first result is shown by considering two concrete diagrams without triple points which present the trivial \(S^2\)-knot, and showing their \(\{\)T1, T2\(\}\)-dependence by using the number of colorings by a set with a quandle-like binary operation satisfying (Q1) and (Q2) but not (Q3) of the three quandle axioms. The second result is shown by constructing two concrete diagrams of an \(S^2\)-knot using the deform-spinning method due to Litherland, and showing their \(\{\)T2\(\}\)-dependence by considering an invariant similar to the quandle cocycle invariant associated with a function \(\theta: Q^3 \to A\) for a quandle \(Q\) and an abelian group \(A\), which satisfies the quandle 3-cocycle condition (i) but not (ii) of the two conditions. In the proof, the authors take \(Q\) as the tetrahedron quandle \(S_4\), and \(A\) as \(\mathbb{Z}\).

MSC:
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57R45 Singularities of differentiable mappings in differential topology
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