rigid isotopy of projective configurations, real algebraic links, Vassiliev invariants.

Jointly with Oleg Viro, we wrote an elementary paper Interlacing of skew lines about configurations of skew lines in the 3-space. It was published in a journal for high school students ``Kvant''. Then we extended this paper to a survey Configuration of skew lines and published it in Leningrad Mathematical Journal in a new rubric ``Light reading for the professional''. We updated it and made an HTML version.

In a paper Linking number in a projective space via the degree of a map, I suggested a construction which expresses the linking number of 1-cycles in the three-dimensional projective space via the degree of a map. For a pair of oriented disjoint circles in , this construction provides a map of a 3-dimensional configuration space to such that its degree is the linking number of the circles multiplied by 2. This is a reasonable replacement of the well-known construction for an affine space. This well-known construction does not work in the projective space. The new construction can be used for representations of Vassiliev invariants and constructions of invariants of real algebraic links.

Recently I found low bounds for the number of lines meeting each of given 4 disjoint smooth closed curves in a given cyclic order in the real projective 3-space and in a given linear order in the Euclidean 3-space. See my preprint Lines and circles joining components of a link, arXiv:math.GT/0511527. Similarly, I estimated the number of circles meeting in a given cyclic order given 6 disjoint smooth closed curves in Euclidean 3-space. The estimations 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 and a surface swept in the 3-space by circles meeting 5 given disjoint smooth closed curves. Higher dimensional generalizations of these results are also outlined.

*Classification of projective Montesinos links*
by Ju. V. Drobotukhina,
*St. Petersburg Math. J.* vol. 3 (1992), No. 1, 97 - 107.

**Abstract:** Links in the three-dimensional projective space
analogous to Montesinos links in the three-dimensional sphere are
classified up to isotopy and up to homeomorphisms.

PDF-file.

*Classification of links in with at most six crossings *
by Julia Drobotukhina,
*Advances in Soviet Mathematics* vol. 18 (1994), No. 1, 87 - 121.

**Abstract:** Links in the three-dimensional projective space
that can be presented by diagrams with at most six crossings are
classified up to isotopy and homeomorphism and tabulated. Reducible and affine links
are excluded from consideration. To obtain a complete list of diagrams,
the approach proposed by Conway for classical links. The main tool for
distinguishing of types is a version of the Jones polynomial.

PDF-file.

*Configurations of skew lines*,
by Julia Viro (Drobotukhina) and Oleg Viro,
Revision (2000) of the paper published in
*Leningrad Math. J.* vol. 1 (1990), No. 4, 1027 - 1050.

**Abstract:** This paper is a survey of results on projective
configurations of subspaces in general position. It is written in the
form of a popular introduction to the subject, with much of the material
accessible to advanced high school students. (As a matter of fact, it
grew up from a popular paper *"Interlacing of skew lines"*
published in a Soviet journal for high-school students * Kvant* in
1988. Here you can find a PDF file with this Russian
paper. Warning: it
is of 11.5 Mb, since it contains color pictures.)
However, in the part of the survey concerning configurations
of lines in general position in three-dimensional space we give a complete
exposition.
Postscript file.

*Linking number in
a projective space via the degree of a map*,
by Julia Viro, Journal of Knot Theorey and Its Ramifications, Vol. 16, No. 4 (2007) 489 - 497. See also
arXiv:math.GT/0405364
[ps, pdf].

** Abstract:** 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.

* Lines joining components of a link*,
by Julia Viro, Journal of Knot Theorey and Its Ramifications, Vol. 18, No. 6 (2009) 865 - 868. See also
arXiv:math.GT/0511527 [ps,
pdf]

**Abstract: ** 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 and in a given linear order in the Euclidean 3-space. Similarly, we estimate the number of circles meeting in a given cyclic order given 6 disjoint smooth closed curves in Euclidean 3-space. The estimations 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 and a surface swept in the 3-space by circles meeting 5 given disjoint smooth closed curves. Higher dimensional generalizations of these results are outlined.