Arf invariants of real algebraic curves

We give some new congruences for singular real algebraic curves which generalize Fiedler's congruence for nonsingular curves.Comment: 14 page

Heegaard genus, cut number, weak p-congruence, and quantum invariants

We use quantum invariants to define a 3-manifold invariant j_p which lies in the non-negative integers. We relate j_p to the Heegard genus, and the cut number. We show that j_$ is an invariant of weak p-congruence.Comment: to appear in JKTR. 8pages 1 figur

Quantum invariants of periodic three-manifolds

Let p be an odd prime and r be relatively prime to p. Let G be a finite p-group. Suppose an oriented 3-manifold M-tilde has a free G-action with orbit space M. We consider certain Witten-Reshetikhin-Turaev SU(2) invariants w_r(M). We will give a fomula for w_r(M) in terms of the defect of M-tilde --> M and the number of elements in G. We also give a version of this result if M and M-tilde contain framed links or colored fat graphs. We give similar formulas for non-free actions which hold for a specified finite set of values for r.Comment: 19 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon2/paper9.abs.htm

Skein theory and Witten-Reshetikhin-Turaev Invariants of links in lens spaces

We study the behavior of the Witten-Reshetikhin-Turaev SU(2) invariants of links in L(p,q) as a function of the level r-2. They are given by 1 over the square root of r times one of p Laurent polynomials evaluated at e to the 2 pi i divided by 4pr. The congruence class of r modulo p determines which polynomial is applicable. If p is zero modulo four, the meridian of L(p,q) is non-trivial in the skein module but has trivial Witten-Reshetikhin-Turaev SU(2) invariants. On the other hand, we show that one may recover the element in the Kauffman bracket skein module of L(p,q) represented by a link from the collection of the WRT invariants at all levels if p is a prime or twice an odd prime. By a more delicate argument, this is also shown to be true for p=9.Comment: Much of the paper has been rewritten and simplified. The only if part of theorem 2 is new. AMS-TeX, 10 page

Floppy Curves with Applications to Real Algebraic Curves

We show how one may sometimes perform singular ambient surgery on the complex locus of a real algebraic curve and obtain what we call a floppy curve. A floppy curve is a certain kind of singular surface in CP(2), more general than the complex locus of a real nodal curve. We derive analogs for floppy curves of known restrictions on real nodal curves. In particular we derive analogs of Fielder's congruence for certain nonsingular curves and Viro's inequalities for nodal curves which generalize those of Arnold and Petrovskii for nonsingular curves. We also obtain a determinant condition for certain curves which are extremal with respect to some of these equalities. One may prohibit certain schemes for real algebraic curves by prohibiting the floppy curves which result from singular ambient surgery. In this way, we give a new proof of Shustin's prohibition of the scheme $1>$ for a real algebraic curve of degree eight.Comment: AmS-TeX- Version 2.1, 38 pages with 16 figures,needs epsf.tex The estimate 9.6 has been improved with corresponding changes in 4.1. The exposition in the proof of 3.4 has been improved. Other minor change

On the Witten-Reshetikhin-Turaev representations of mapping class groups

We consider a central extension of the mapping class group of a surface with a collection of framed colored points. The Witten-Reshetikhin-Turaev TQFTs associated to SU(2) and SO(3) induce linear representations of this group. We show that the denominators of matrices which describe these representation over a cyclotomic field can be restricted in many cases. In this way, we give a proof of the known result that if the surface is a torus and there are no colored points, the representations have finite image.Comment: AMS-TeX, 7 pages. Notational changes and typos corrected. To appear in Proc. A.M.