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Rail knotoids
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abstract
We work on the notions of rail arcs and rail isotopy in $\mathbb{R}^3$, and we introduce the notions of rail knotoid diagrams and their equivalence. Our main result is that two rail arcs in $\mathbb{R}^3$ are rail isotopic if and only if their knotoid diagram projections onto the plane of the two lines which we call rails, are equivalent. We also make a connection between the rail isotopy in $\mathbb{R}^3$ and the knot theory of the handlebody of genus $2$.
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Braidoids
Braidoids are defined and shown to satisfy analogues of the Alexander and Markov theorems, so planar knotoids can be braided and braidoid equivalence corresponds to knotoid isotopy.
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