arXiv subjects
Publications and source records attributed to Patrick M. Gilmer.
Let Sigma be a closed oriented surface of genus g. We show that the Kauffman bracket skein module of Sigma x S^1 over the field of rational functions in A has dimension at least 2^{2g+1}+2g-1.
We define and calculate signature and nullity invariants for complex schemes for curves in the real projective plane. We use an analog of the Murasugi-Tristram inequality to prohibit certain schemes from being realized by real algebraic curves. We give new formulas for Casson-Gordon invariants of graph manifolds, and signatures of graph links.
For p>3 a prime, and g>2 an integer, we use Topological Quantum Field Theory (TQFT) to study a family of p-1 highest weight modules L_p(lambda) for the symplectic group Sp(2g,K) where K is an algebraically closed field of characteristic p. This permits explicit formulae for the dimension and the formal character of L_p(lambda) for these highest weights.
This paper refines previous work by the first author. We study the question of which links in the 3-sphere can be obtained as closures of a given 1-manifold in an unknotted solid torus in the 3-sphere (or genus-1 tangle) by adjoining another 1-manifold in the complementary solid torus. We distinguish between even and odd closures, and define even and odd versions of the Kauffman bracket ideal. These even and odd Kauffman bracket ideals are used to obstruct even and odd tangle closures. Using a basis of Habiro's for the even Kauffman bracket skein module of the solid torus, we define bases for the even and odd skein module of the solid torus relative to two points. These even and odd bases allow us to compute a finite list of generators for the even and odd Kauffman bracket ideals of a genus-1 tangle. We do this explicitly for three examples. Furthermore, we use the even and odd Kauffman bracket ideals to conclude in some cases that the determinants of all even/odd closures of a genus-1 tangle possess a certain divisibility.
Carrega has shown that the Kauffman bracket skein module of the 3-torus over the field of rational functions in the variable A can be generated by 9 skein elements. We show this set of generators is linearly independent.
We relate the jumps of the signature function of a link to the roots of its first nonzero higher Alexander polynomial.
We consider two functions on Sp(g,R) with values in the cyclic group of order four {1,-1,i,-i}. One was defined by Lion and Vergne. The other is -i raised to the power given by an integer valued function defined by Masbaum and the author (initially on the mapping class group of a surface). We identify these functions when restricted to Sp(g,Z). We conjecture the identity of these functions on Sp(g,R).
If a knot K bounds a genus one Seifert surface F in the 3-sphere and F contains an essential simple closed curve alpha that has induced framing 0 and is smoothly slice, then K is smoothly slice. Conjecturally, the converse holds. It is known that if K is slice, then there are strong constraints on the algebraic concordance class of such alpha, and it was thought that these constraints might imply that alpha is at least algebraically slice. We present a counterexample; in the process we answer negatively a question of Cooper and relate the result to a problem of Kauffman. Results of this paper depend on the interplay between the Casson-Gordon invariants of K and algebraic invariants of alpha.
We find decomposition series of length at most two for modular representations in positive characteristic of mapping class groups of surfaces induced by an integral version of the Witten-Reshetikhin-Turaev SO(3)-TQFT at the p-th root of unity, where p is an odd prime. The dimensions of the irreducible factors are given by Verlinde-type formulas.
By applying a variant of the TQFT constructed by Blanchet, Habegger, Masbaum, and Vogel, and using a construction of Ohtsuki, we define a module endomorphism for each knot K by using a tangle obtained from a surgery presentation of K. We show that it is strong shift equivalent to the Turaev-Viro endomorphism associated to K. Following Viro, we consider the endomorphisms that one obtains after coloring the meridian and longitude of the knot. We show that the traces of these endomorphisms encode the same information as the colored Jones polynomials of K at a root of unity. Most of the discussion is carried out in the more general setting of infinite cyclic covers of 3-manifolds.
A TQFT is a functor from a cobordism category to the category of vector spaces, satisfying certain properties. An important property is that the vector spaces should be finite dimensional. For the WRT TQFT, the relevant 2+1-cobordism category is built from manifolds which are equipped with an extra structure such as a p_1-structure, or an extended manifold structure. We perform the universal construction of Blanchet, Habegger, Masbaum and Vogel on a cobordism category without this extra structure and show that the resulting quantization functor assigns an infinite dimensional vector space to the torus.
We compare the dimensions of the irreducible Sp(2g,K)-modules over a field K of characteristic p constructed by Gow with the dimensions of the irreducible Sp(2g,F_p)-modules that appear in the first approximation to representations of mapping class groups of surfaces in Integral Topological Quantum Field Theory. For this purpose, we derive a trigonometric formula for the dimensions of Gow's representations. This formula is equivalent to a special case of a formula contained in unpublished work of Foulle. Our direct proof is simpler than the proof of Foulle's more general result, and is modeled on the proof of the Verlinde formula in TQFT.
Given a mapping class f of an oriented surface Sigma and a lagrangian lambda in the first homology of Sigma, we define an integer n_{lambda}(f). We use n_{lambda}(f) (mod 4) to describe a universal central extension of the mapping class group of Sigma as an index-four subgroup of the extension constructed from the Maslov index of triples of lagrangian subspaces in the homology of the surface. We give two descriptions of this subgroup. One is topological using surgery, the other is homological and builds on work of Turaev and work of Walker. Some applications to TQFT are discussed. They are based on the fact that our construction allows one to precisely describe how the phase factors that arise in the skein theory approach to TQFT-representations of the mapping class group depend on the choice of a lagrangian on the surface.
We give new explicit formulas for the representations of the mapping class group of a genus one surface with one boundary component which arise from Integral TQFT. Our formulas allow one to compute the h-adic expansion of the TQFT-matrix associated to a mapping class in a straightforward way. Truncating the h-adic expansion gives an approximation of the representation by representations into finite groups. As a special case, we study the induced representations over finite fields and identify them up to isomorphism. The key technical ingredient of the paper are new bases of the Integral TQFT modules which are orthogonal with respect to the Hopf pairing. We construct these orthogonal bases in arbitrary genus, and briefly describe some other applications of them.
We develop obstructions to a knot K in the 3-sphere bounding a smooth punctured Klein bottle in the 4-ball. The simplest of these is based on the linking form of the 2-fold branched cover of the 3-sphere branched over K. Stronger obstructions are based on the Ozsvath-Szabo correction term in Heegaard-Floer homology, along with the G-signature theorem and the Guillou-Marin generalization of Rokhlin's theorem. We also apply Casson-Gordon theory to show that for every n greater than one there exists a knot that does not bound a topologically embedded nonorientable ribbon surface F in the 4-ball with first Betti number less than n.
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.
We give a congruence for the quantum invariant of a Z_p-quotient of a 3$-manifold with a Z_{p^2} action. We show the congruence does not hold for quotients of 3--manifolds with a Z_{5}xZ_{5} action.
Let f be an integer greater than one. We study three progressively finer equivalence relations on closed 3-manifolds generated by Dehn surgery with denominator f: weak f-congruence, f-congruence, and strong f-congruence. If f is odd, weak f-congruence preserves the ring structure on cohomology with Z_f-coefficients. We show that strong f-congruence coincides with a relation previously studied by Lackenby. Lackenby showed that the quantum SU(2) invariants are well-behaved under this congruence. We strengthen this result and extend it to the SO(3) quantum invariants. We also obtain some corresponding results for the coarser equivalence relations, and for quantum invariants associated to more general modular categories. We compare S^3, the Poincare homology sphere, the Brieskorn homology sphere Sigma(2,3,7) and their mirror images up to strong f-congruence. We distinguish the weak f-congruence classes of some manifolds with the same Z_f-cohomology ring structure.