53 citations · 291 across the 36 of their papers we have counts for
38 papers
On the skein module of the product of a surface and a circle
Patrick M. Gilmer, Gregor Masbaum
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^…
An application of TQFT to modular representation theory
Patrick M. Gilmer, Gregor Masbaum
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) wh…
On the Kauffman bracket skein module of the 3-torus
Patrick M. Gilmer
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 se…
Signatures of real algebraic curves via plumbing diagrams
Patrick M. Gilmer, Stepan Yu. Orevkov
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 prohib…
Signature Jumps and Alexander Polynomials for Links
Patrick M. Gilmer, Charles Livingston
We relate the jumps of the signature function of a link to the roots of its first nonzero higher Alexander polynomial.
Even and odd Kauffman bracket ideals for genus-1 tangles
Susan M. Abernathy, Patrick M. Gilmer
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 t…
Two functions on Sp(g,R)
Patrick M. Gilmer
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…
On the Turaev-Viro endomorphism, and the colored Jones polynomial
Xuanting Cai, Patrick M. Gilmer
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 usi…
Extra structure and the universal construction for the Witten-Reshetikhin-Turaev TQFT
Patrick M. Gilmer, Xuanye Wang
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 dime…
Dimension formulas for some modular representations of the symplectic group in the natural characteristic
Patrick M. Gilmer, Gregor Masbaum
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 a…
Irreducible factors of modular representations of mapping class groups arising in Integral TQFT
Patrick M. Gilmer, Gregor Masbaum
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…
On surgery curves for genus one slice knots
Patrick M. Gilmer, Charles Livingston
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 s…
The nonorientable four-genus of knots
Patrick M. Gilmer, Charles Livingston
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 branche…
Maslov index, Lagrangians, Mapping Class Groups and TQFT
Patrick M. Gilmer, Gregor Masbaum
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 des…
Integral TQFT for a one-holed torus
Patrick M. Gilmer, Gregor Masbaum
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 all…
Heegaard genus, cut number, weak p-congruence, and quantum invariants
Patrick M. Gilmer
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…
Remarks on Congruence of 3-manifolds
Patrick M. Gilmer
We give two proofs that the 3-torus is not weakly d-congruent to the connected sum of three S^1xS^2's, if d>2. We study how cohomology ring structure relates to weak congruence. We…
Quantum invariants and free Z_{p^2} -actions on 3-manifolds
Patrick M. Gilmer, Khaled Qazaqzeh
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…
Congruence and similarity of 3-manifolds
Patrick M. Gilmer
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…
Arf invariants of real algebraic curves
Patrick M. Gilmer
We give some new congruences for singular real algebraic curves which generalize Fiedler's congruence for nonsingular curves.
On the Frohman Kania-Bartoszynska ideal
Patrick M. Gilmer
The Frohman Kania-Bartoszynska ideal is an invariant associated to a 3-manifold with boundary and a prime p >3. We give some estimates of this ideal. We also calculate this invaria…
Integral Lattices in TQFT
Patrick M. Gilmer, Gregor Masbaum
We find explicit bases for naturally defined lattices over a ring of algebraic integers in the SO(3) TQFT-modules of surfaces at roots of unity of odd prime order. Some application…
On the Kauffman bracket skein module of the quaternionic manifold
Patrick M. Gilmer, John M. Harris
We use recoupling theory to study the Kauffman bracket skein module of the quaternionic manifold over Z[A,A^{-1}] localized by inverting all the cyclotomic polynomials. We prove th…
The parity of the Maslov index and the even cobordism category
Patrick M. Gilmer, Khaled Qazaqzeh
We give a formula for the parity of the Maslov index of a triple of Lagrangian subspaces of a skew symmetric bilinear form over the real numbers. We define an index two subcategory…