Julia Viro,   research

Research interests

Low-dimensional topology and knot theory, links in the projective space,
rigid isotopy of projective configurations, real algebraic links, Vassiliev invariants.

Overview of research results

In my first research papers I studied links in the three-dimensional real projective space $ \mathbb{R}P^3$. In paper An analogue of the Jones polynomial for links in $ \mathbb{R}P^3$ and generalization of the Kauffman-Murasugi theorem I discussed the notion of a diagram for a link in $ \mathbb{R}P^3$ and generalized the notion of alternating diagrams to projective links. I extended the Kauffman bracket link polynomial to projective links and generalized the Kauffman-Murasugi theorems on characterization of alternating links in terms of the Jones polynomial to the case of projective links. In the next two papers, Classification of projective Montesinos links and Classification of links in $ \mathbb{R}P^3$ with at most six crossings, I classified projective links belonging to few simple classes up to homeomorphism and ambient isotopy.

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 $ \mathbb{R}P^3$, this construction provides a map of a 3-dimensional configuration space to $ \mathbb{R}P^3$ 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.

Publications

An analogue of the Jones polynomial for links in $ \mathbb{R}P^3$ and generalization of the Kauffman-Murasugi theorem by Ju. V. Drobotukhina, Leningrad Math. J. vol. 2 (1991), No. 3, 613 - 630.
Abstract: We define analogoues of the Jones polynomial for links in the projective space $ \mathbb{R}P^3$. We prove corresponding generalizations of the Kauffman - Murasugi theorem on the connection between the combinatorial properties of a link diagram and the properties of the Jones polynomial. Finally, we study criteria for isotopy of a link in the space $ \mathbb{R}P^3$ to a link lying in an affine part of the space. PDF-file.

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 $ \mathbb{R}P^3$ 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.