arXiv subjects

D. N. Yetter

Publications and source records attributed to D. N. Yetter.

16 recordsLinked to original sources

arXiv2023

Complementation of Subquandles

Saki and Kiani proved that the subrack lattice of a rack $R$ is necessarily complemented if $R$ is finite but not necessarily complemented if $R$ is infinite. In this paper, we investigate further avenues related to the complementation of subquandles. Saki and Kiani's example of an infinite rack without complements is a quandle, which is neither ind-finite nor profinite. We provide an example of an ind-finite quandle whose subobject lattice is not complemented, and conjecture that profinite quandles have complemented subobject lattices. Additionally, we provide a complete classification of subquandles whose set-theoretic complement is also a subquandle, which we call \textit{strongly complemented}, and provide a partial transitivity criterion for the complementation in chains of strongly complemented subquandles. One technical lemma used in establishing this is of independent interest: the inner automorphism group of a subquandle is always a subquotient of the inner automorphism group of the ambient quandle.

math.GR↗

arXiv2020

Bicategories for TQFTs with Defects with Structure

We provide a description of adequate categorical data to give a Turaev-Viro type state-sum construct of invariants of 3-manifolds with a system of defects, generalizing the Dijkgraaf-Witten type invariants of our earlier work. We term the defects in our construction defects-with-structure because algebraic data associated to them is in general richer than a module category over the spherical fusion category from which the theory is constructed when no defect is present.

math.QA↗

arXiv2019

Factorizations of surjective maps of connected quandles

We consider the problem of when one quandle homomorphism will factor through another, restricting our attention to the case where all quandles involved are connected. We provide a complete solution to the problem for surjective quandle homomorphisms using the structure theorem for connected quandles of Ehrman et al. (2008) and the factorization system for surjective quandle homomorphsims of Bunch et al. (2010) as our primary tools. The paper contains the substantive results obtained by an REU research group consisting of the first four authors under the mentorship of the fifth, and was supported by National Science Foundation, grant DMS-1659123.

math.GR↗

arXiv2018

Multi-Skein Invariants for Welded and Extended Welded Knots and Links

The theory of welded and extended welded knots is a generalization of classical knot theory. Welded (resp. extended welded) knot diagrams include virtual crossings (resp. virtual crossings and wen marks) and are equivalent under an extended set of Reidemeister-type moves. We present a new class of invariants for welded and extended welded knots and links using a multi-skein relation, following Z. Yang's approach for virtual knots. Using this skein-theoretic approach, we find sufficient conditions on the coefficients to obtain invariance under the extended Reidemeister moves appropriate to welded and extended welded links.

math.GT↗

arXiv2018

Stratified spaces, Directed Algebraic Topology, and State-Sum TQFTs

We apply the theory of directed topology developed by Grandis [9, 10] to the study of stratified spaces by describing several ways in which a stratification or a stratification with orientations on the strata can be used to produce a related directed space structure. This description provides a setting for the constructions of state-sum TQFTs with defects of [5, 8], which we extend to a similar construction of a Dijkgraaf- Witten type TQFT in the case where the defects (lower dimensional strata) are not sources or targets, but sources on one side and targets on the other, according to an orientation convention.

math.AT↗

arXiv2015

Dijkgraaf-Witten Type Invariants of Seifert Surfaces in 3-Manifolds

We introduce defects, with internal gauge symmetries, on a knot and Seifert surface to a knot into the combinatorial construction of finite gauge-group Dijkgraaf-Witten theory. The appropriate initial data for the construction are certain three object categories, with coefficients satisfying a partially degenerate cocycle condition.

math.QA↗

arXiv2013

On Deformations of Pasting Diagrams, II

We continue the development of the infinitesimal deformation theory of pasting diagrams of k-linear categories begun in Yetter, D.N. "On Deformations of Pasting Diagrams", Theory and Applications of Categories 22 (2009) 24-53. In that paper, the standard result that all obstructions are cocycles was established only for the elementary, composition-free parts of pasting diagrams. In the present work we give a proof for pasting diagrams in general. As tools we use (1) the method developed by Shrestha, in his Kansas State University doctoral dissertation, of representing formulas for obstructions, along with the corresponding cocycle and cobounding conditions by suitably labeled polygons, giving a rigorous exposition of the previously heuristic method, and (2) deformations of pasting diagrams in which some cells are required to be deformed trivially.

math.CT↗

arXiv2009

On Deformations of Pasting Diagrams

We adapt the work of Power to describe general, not-necessarily composable, not-necessarily commutative 2-categorical pasting diagrams and their composable and commutative parts. We provide a deformation theory for pasting diagrams valued in $k$-linear categories, paralleling that provided for diagrams of algebras by Gerstenhaber and Schack, proving the standard results. Along the way, the construction gives rise to a bicategorical analog of the homotopy G-algebras of Gerstenhaber and Voronov.

math.CT↗

arXiv2008

On a Basis for the Framed Link Vector Space Spanned by Chord Diagrams

In view of the result of Kontsevich, now often called ``the fundamental theorem of Vassiliev theory'', identifying the graded dual of the associated graded vector space to the space of Vassiliev invariants filtered by degree with the linear span of chord diagrams modulo the ``4T-relation'' (and in the unframed case, the ``1T-'' or ``isolated chord relation''), it is a problem of some interest to provide a basis for the space of chord diagrams modulo the 4T-relation. We construct the basis for the vector space spanned by chord diagrams with n chords and m distinguishable link components, modulo 4T relations for n less than or equal to 5.

math.GT↗

arXiv2005

Toward a classification of finite quandles

This paper summarizes substantive new results derived by a student team (the first three authors) under the direction of the fourth author at the 2005 session of the KSU REU ``Brainstorming and Barnstorming''. The main results are a decomposition theorem for quandles in terms of an operation of `semidisjoint union' showing that all finite quandles canonically decompose via iterated semidisjoint unions into connected subquandles, and a structure theorem for finite connected quandles with prescribe inner automorphism group. The latter theorem suggests a new approach to the classification of finite connected quandles.

math.QA↗

arXiv2005

Invariants of smooth 4-manifolds via Vassiliev theory

The above named paper has been withdrawn. A colleague has observed a gap in the proof of isotopy invariance, which can be repaired by reducing the coefficients (which lie in (1/6)Z) of the antisymmetric kanji with chords incident with more than one component modulo 8Z. An analogous issue arises in considering the effect of orientation reversal in the proof of handle-slide invariance, which can be repaired by reducing the coefficients of all of the antisymmetric kanji modulo 4Z. While an invariant of smooth 4-manifolds is obtained by the repaired construction, the corrected construction does not distinguish the smooth structures on the non-diffeomorphic but homeomorphic pairs of Gompf nuclei. A more subtle approach to restoring invariance, using an action of an extension of Gl(n,Z), rather than quotienting the coefficient, has been examined, but will also not restore the ability of the invariant constructed to distinguish the Gompf nuclei.

math.GT↗

arXiv2004

Measurable Categories

We develop the theory of categories of measurable fields of Hilbert spaces and bounded fields of bounded operators. We examine classes of functors and natural transformations with good measure theoretic properties, providing in the end a rigorous construction for the bicategory used in joint work with Crane as the basis for a representation theory of (Lie) 2-groups. Several important technical results are established along the way: First it is shown that all bounded invertible additive functors (and thus a fortiori all invertible *-functors) between categories of measurable fields of Hilbert spaces are induced by invertible measurable transformations between the underlying Borel spaces. Second the distributivity of Hilbert space tensor product over direct integrals over Lusin spaces with respect to $σ$-finite measures is established. The paper concludes with a general definition of measurable bicategories.

math.CT↗

arXiv2003

Measurable Categories and 2-Groups

Using the theory of measurable categories developped by Yetter in work in preparation, we provide a notion of representations of 2-groups more well-suited to physically and geometrically interesting examples than that proposed in unpublished work of Barrett and Mackaay using 2-VECT of Kapranov and Voevodsky. Using this theory we sketch a 2-categorical approach to the state-sum model for Lorentzian quantum gravity proposed in by the authors in previous work and suggest state-integral constructions for 4-manifold invariants.

math.QA↗

arXiv2002

Quandles and Monodromy

We show that a variety of monodromy phenomena arising in geometric topology and algebraic geometry are most conveniently described in terms of quandle homomorphisms from a knot quandle associated to the base to a quandle associated to a fiber. We consider the cases of the monodromy of a branched covering, braid monodromy and the monodromy of a Lefschetz fibration.

math.GT↗

arXiv2002

Quandles and Lefschetz Fibrations

We show that isotopy classes of simple closed curves in any oriented surface admit a quandle structure with operations induced by Dehn twists, the Dehn quandle of the surface. We further show that the monodromy of a Lefschetz fibration can be conveniently encoded as a quandle homomorphism from the knot quandle of the base as a manifold with a codimension 2 subspace (the set of singular values) to the Dehn quandle of the generic fibre, and discuss prospects for construction of invariants arising naturally from this description of the monodromy.

math.GT↗

arXiv1997

On the Classical Limit of the Balanced State Sum

The purpose of this note is to make several advances in the interpretation of the balanced state sum model by Barrett and Crane in gr-qc/9709028 as a quantum theory of gravity. First, we outline a shortcoming of the definition of the model in pointed out to us by Barrett and Baez in private communication, and explain how to correct it. Second, we show that the classical limit of our state sum reproduces the Einstein-Hilbert lagrangian whenever the term in the state sum to which it is applied has a geometrical interpretation. Next we outline a program to demonstrate that the classical limit of the state sum is in fact dominated by terms with geometrical meaning. This uses in an essential way the alteration we have made to the model in order to fix the shortcoming discussed in the first section. Finally, we make a brief discussion of the Minkowski signature version of the model.

gr-qc↗