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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
##### Keywords:
surface-link; diagram; Roseman move; $$S$$-dependence
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