Narrow Results

For a smooth, irreducible projective surface S over ℂ, the number of r-nodal curves in an ample linear system (where is a line bundle on S) can be expressed using the rth Bell polynomial P_r in universal functions a_i, 1 ≤ i ≤ r, of (S, ), which are ℤ-linear polynomials in the four Chern numbers of S and . We use this result to establish a proof of the classical shape conjectures of Di Francesco–Itzykson and Göttsche governing node polynomials in the case of ℙ². We also give a recursive procedure which provides the -term of the polynomials a_i.

A family of sets is said to have property B(s) if there is a set, referred to as a blocking set, whose intersection with each member of the family is a proper subset of that blocking set and contains fewer than s elements. A finite projective plane is a construction satisfying the two conditions that any two lines meet in a unique point and any two points are on a unique line. In this paper, the authors develop an algorithm of complexity O(n³) for constructing a blocking set for a projective plane of order n.

We study Vassiliev invariants of links in a 3-manifold M by using chord diagrams labeled by elements of the fundamental group of M. We construct universal Vassiliev invariants of links in M, where M=P²×[0,1] is a cylinder over the real projective plane P², M=Σ×[0,1] is a cylinder over a surface Σ with boundary, and M=S¹×S². A finite covering p:N→M induces a map π₁(p)* between labeled chord diagrams that corresponds to taking the preimage p^-1(L)⊂N of a link L⊂M. The maps p^-1 and π₁(p)* intertwine the constructed universal Vassiliev invariants.

This paper is a survey paper that gives detailed constructions and illustrations of some of the standard examples of non-orientable surfaces that are embedded and immersed in 4-dimensional space. The illustrations depend upon their 3-dimensional projections, and indeed the illustrations here depend upon a further projection into the plane of the page. The concepts used to develop the illustrations will be developed herein.

In the present paper, we proceed with the study of framed 4-graph minor theory initiated in [V. O. Manturov, Framed 4-valent graph minor theory I: Intoduction planarity criterion, arxiv: 1402.1564v1 [Math.Co]] and justify the planarity theorem for arbitrary framed 4-graphs; besides, we prove analogous results for embeddability in $ℝP^2$.