Consider two parallel lines [Formula: see text] and [Formula: see text] in [Formula: see text]. A rail arc is an embedding of an arc in [Formula: see text] such that one endpoint is on [Formula: see text], the other is on [Formula: see text], and its interior is disjoint from [Formula: see text]. Rail arcs are considered up to rail isotopies, ambient isotopies of [Formula: see text] with each self-homeomorphism mapping [Formula: see text] and [Formula: see text] onto themselves. When the manifolds and maps are taken in the piecewise linear category, these rail arcs are called stick rail arcs. The stick number of a rail arc class is the minimum number of sticks, line segments in a p.l. arc, needed to create a representative. This paper calculates the stick number of rail arcs classes with a crossing number at most 2 and uses a winding number invariant to calculate the stick numbers of infinitely many rail arc classes. Each rail arc class has two canonically associated knot classes, its under and over companions. This paper also introduces the rail stick number of knot classes, the minimum number of sticks needed to create a rail arcs whose under or over companion is the knot class. The rail stick number is calculated for 29 knot classes with crossing number at most 9. The stick number of multi-component rail arcs classes is considered as well as the lattice stick number of rail arcs.
The axioms of a quandle imply that the columns of its Cayley table are permutations. This paper studies quandles with exactly one non-trivially permuted column. Their automorphism groups, quandle polynomials, (symmetric) cohomology groups, and Hom quandles are studied. The quiver and cocycle invariant of links using these quandles are shown to relate to linking number.
Yoshikawa made a table of knotted surfaces in [Formula: see text] with ch-index 10 or less. This remarkable table is the first to enumerate knotted surfaces analogous to the classical prime knot table. A broken sheet diagram of a surface-link is a generic projection of the surface in [Formula: see text] with crossing information along its singular set. The minimal number of triple points among all broken sheet diagrams representing a given surface-link is its triple point number. This paper compiles the known triple point numbers of the surface-links represented in Yoshikawa’s table and calculates or provides bounds on the triple point number of the remaining surface-links.
The vertex distortion of a lattice knot is the supremum of the ratio of the distance between a pair of vertices along the knot and their distance in the [Formula: see text]-norm. Inspired by Gromov, Pardon and Blair–Campisi–Taylor–Tomova, we show that results about the distortion of smooth knots hold for vertex distortion: the vertex distortion of a lattice knot is 1 only if it is the unknot, and there are minimal lattice-stick number knot conformations with arbitrarily high distortion.
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