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For any two disjoint oriented circles embedded into the 3-dimensional real projective space, we construct a 3-dimensional configuration space and its map to the projective space such that the linking number of the circles is the half of the degree of the map. Similar interpretations are given for the linking number of cycles in a projective space of arbitrary odd dimension and the self-linking number of a zero homologous knot in the 3-dimensional projective space.

Homology of the circle with non-trivial local coefficients is trivial. From this well-known fact we deduce geometric corollaries involving codimension-two links. In particular, the Murasugi–Tristram signatures are extended to invariants of links formed of arbitrary oriented closed codimension two submanifolds of an odd-dimensional sphere. The novelty is that the submanifolds are not assumed to be disjoint, but are transversal to each other, and the signatures are parametrized by points of the whole torus. Murasugi–Tristram inequalities and their generalizations are also extended to this setup.

We estimate from below the number of lines meeting each of given 4 disjoint smooth closed curves in a given cyclic order in the real projective 3-space. We obtain also a similar lower bound for the number of lines meeting each of given 4 disjoint smooth closed curves in a given linear order in ℝ³. The estimates are formulated in terms of linking numbers of the curves and obtained by orienting of the corresponding configuration spaces and evaluating of their signatures. This involves a study of a surface swept by lines meeting 3 given disjoint smooth closed curves. Higher-dimensional generalizations of these results are outlined.

The core group of a classical link was introduced independently by Kelly in 1991 and Wada in 1992. It is a link invariant defined by a presentation involving the arcs and crossings of a diagram, related to Wirtinger’s presentation of the fundamental group of a link complement. Two close relatives of the core group are defined by presentations involving regions rather than arcs; one of them is related to Dehn’s presentation of a link group. The definitions are extended to virtual link diagrams and properties of the resulting invariants are discussed.

In this paper, we answer a question raised in “Peripheral elements in reduced Alexander modules” [J. Knot Theory Ramifications 31 (2022) 2250058]. We also correct a minor error in that paper.

We give a geometric description of welded links in the spirit of Kuperberg's description of virtual links: “What is a virtual link?” [8].