# Efstratia KalfagianniMichigan State University | MSU · Department of Mathematics

Efstratia Kalfagianni

Professor

## About

108

Publications

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Introduction

## Publications

Publications (108)

We present a coarse perspective on relations of the SU (2)-Witten-Reshetikhin-Turaev TQFT, the Weil-Petersson geometry of the Teichmüller space, and volumes of hy-perbolic 3-manifolds. Using data from the asymptotic expansions of the curve operators in the skein theoretic version of the SU (2)-TQFT, as developed by Blanchet, Habegger, Masbaum and V...

The proof of Witten's finiteness conjecture established that the Kauffman bracket skein modules of closed $3$-manifolds are finitely generated over $\Q(A)$. In this paper, we develop a novel method for computing these skein modules.
We show that if the skein module $S(M,\Q[A^\pmo])$ of $M$ is tame (e.g. finitely generated over $\Q[A^{\pm 1}]$), a...

We use the degree of the colored Jones knot polynomials to show that the crossing number of a $(p,q)$-cable of an adequate knot with crossing number $c$ is larger than $q^2\, c$. As an application we determine the crossing number of $2$-cables of adequate knots.

We use Dehn surgery methods to construct infinite families of hyperbolic knots in the 3-sphere satisfying a weak form of the Turaev-Viro invariants volume conjecture. The results have applications to a conjecture of Andersen, Masbaum, and Ueno about quantum representations of surface mapping class groups. We obtain an explicit family of pseudo-Anos...

We use Dehn surgery methods to construct infinite families of hyperbolic knots in the 3-sphere satisfying a weak form of the Turaev--Viro invariants volume conjecture. The results have applications to a conjecture of Andersen, Masbaum, and Ueno about quantum representations of surface mapping class groups. We obtain an explicit family of pseudo-Ano...

It has long been known that the quadratic term in the degree of the colored Jones polynomial of a knot is bounded above in terms of the crossing number of the knot. We show that this bound is sharp if and only if the knot is adequate. As an application of our result we determine the crossing numbers of broad families of non-adequate prime satellite...

We prove that for any closed, connected, oriented 3-manifold M , there exists an infinite family of 2-fold branched covers of M that are hyperbolic 3-manifolds and surface bundles over the circle with arbitrarily large volume.

We prove that for any closed, connected, oriented 3-manifold M, there exists an infinite family of 2-fold branched covers of M that are hyperbolic 3-manifolds and surface bundles over the circle with arbitrarily large volume.

Weakly generalised alternating knots are knots with an alternating projection onto a closed surface in a compact irreducible 3-manifold, and they share many hyperbolic geometric properties with usual alternating knots. For example, usual alternating knots have volume bounded above and below by the twist number of the alternating diagram due to Lack...

We use geometric methods to show that given any 3-manifold M , and g a sufficiently large integer, the mapping class group Mod(Σ g,1) contains a coset of an abelian subgroup of rank g 2 , consisting of pseudo-Anosov monodromies of open-book decompositions in M. We prove a similar result for rank two free cosets of Mod(Σ g,1). These results have app...

It has long been known that the quadratic term in the degree of the colored Jones polynomial of a knot is bounded above in terms of the crossing number of the knot. We show that this bound is sharp if and only if the knot is adequate. As an application of our result we determine the crossing numbers of broad families of non-adequate prime satellite...

We show that the strong slope conjecture implies that the degrees of the colored Jones knot polynomials detect the figure 8 knot. Furthermore, we propose a characterization of alternating knots in terms of the Jones period and the degree span of the colored Jones polynomial

We consider hyperbolic links that admit alternating projections on surfaces in compact, irreducible 3-manifolds. We show that, under some mild hypotheses, the volume of the complement of such a link is bounded below in terms of a Kauffman bracket function defined on link diagrams on the surface. In the case that the 3-manifold is a thickened surfac...

Weakly generalised alternating knots are knots with an alternating projection onto a closed surface in a compact irreducible 3-manifold, and they share many hyperbolic geometric properties with usual alternating knots. For example, usual alternating knots have volume bounded above and below by the twist number of the alternating diagram due to Lack...

We point out that the strong slope conjecture implies that the degrees of the colored Jones knot polynomials detect the figure eight knot. Furthermore, we propose a characterization of alternating knots in terms of the Jones period and the degree span of the colored Jones polynomial.

We use geometric methods to study a conjecture of Andersen, Masbaum and Ueno about quantum representations of surface mapping class groups. For surfaces with boundary, and large enough genus, we construct cosets of abelian and free subgroups of their mapping class groups consisting of elements that satisfy the conjecture. The mapping tori of these...

We survey some tools and techniques for determining geometric properties of a link complement from a link diagram. In particular, we survey the tools used to estimate geometric invariants in terms of basic diagrammatic link invariants. We focus on determining when a link is hyperbolic, estimating its volume, and bounding its cusp shape and cusp are...

We show that the strong slope conjecture implies that the degree of the colored Jones polynomial detects all torus knots. As an application we obtain that an adequate knot that has the same colored Jones polynomial degrees as a torus knot must be a (2, q)-torus knot.

In this talk I will discuss recent progress on understanding the asymptotic behavior of certain representations of surface mapping class groups. I will also discuss
some geometric properties of surface bundles detected by these asymptotics. The talk is based on joint work with Renaud Detcherry and partly with Giulio Belletti and Tian

We observe that the strong slope conjecture implies that the degree of the colored Jones polynomial detects all torus knots. As an application we obtain that an adequate knot that has the same colored Jones polynomial degrees as a torus knot must be a (2, q)-torus knot.

We observe that the strong slope conjecture implies that the degree of the colored Jones polynomial detects all torus knots. As an application we obtain that an adequate knot that has the same colored Jones polynomial degrees as a torus knot must be a $(2,q)$-torus knot.

We prove the Turaev-Viro invariants volume conjecture for complements of fundamental shadow links: an infinite family of hyperbolic link complements in connected sums of copies of $S^1\times S^2$. The main step of the proof is to find a sharp upper bound on the growth rate of the quantum $6j-$symbol evaluated at $e^{\frac{2\pi i}{r}}.$ As an applic...

We prove the Turaev-Viro invariants volume conjecture for complements of fundamental shadow links: an infinite family of hyperbolic link complements in connected sums of copies of S 1 × S 2. The main step of the proof is to find a sharp upper bound on the growth rate of the quantum 6j−symbol evaluated at e 2πi r. As an application of the main resul...

State surfaces are spanning surfaces of links that are obtained from link diagrams guided by the combinatorics underlying Kauffman's construction of the Jones polynomial via state models. Geometric properties of such surfaces are often dictated by simple link diagrammatic criteria, and the surfaces themselves carry important information about geome...

Around 1990 Turaev and Viro defined a family of real-valued 3-manifold invariants as state sums on triangulations of 3-manifolds. The family is indexed by an integer (the level) and at each level the invariant depends on a certain root of unity. We will discuss recent results making progress in understanding the `large-level” asymptotic behavior of...

We establish a relation between the
``large r" asymptotics of the Turaev-Viro invariants $TV_r$ and the Gromov norm of 3-manifolds.
We show that for any orientable, compact 3-manifold $M$, with (possibly empty) toroidal boundary, $\log |TV_r (M)|$ is bounded above by a function linear in $r$ and whose slope is
a positive universal constant times t...

Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level $r$). In this paper, we relate the AMU conjecture to a question about the growth of the Turaev-Viro invariants $TV_r$ of hyperbolic 3-manifolds. W...

We survey some tools and techniques for determining geometric properties of a link complement from a link diagram. In particular, we survey the tools used to estimate geometric invariants in terms of basic diagrammatic link invariants. We focus on determining when a link is hyperbolic, estimating its volume, and bounding its cusp shape and cusp are...

Geometric structures of 3-manifolds and quantum invariants

We describe a normal surface algorithm that decides whether a knot, with known degree of the colored Jones polynomial, satisfies the Strong Slope Conjecture. We also discuss possible simplifications of our algorithm and state related open questions. We establish a relation between the Jones period of a knot and the number of sheets of the surfaces...

We obtain a formula for the Turaev-Viro invariants of a link complement in terms of values of the colored Jones polynomial of the link. As an application we give the first examples for which the volume conjecture of Chen and the third named author\,\cite{Chen-Yang} is verified. Namely, we show that the asymptotics of the Turaev-Viro invariants of t...

We derive bounds on the length of the meridian and the cusp volume of hyperbolic knots in terms of the topology of essential surfaces spanned by the knot. We provide an algorithmically checkable criterion that guarantees that the meridian length of a hyperbolic knot is below a given bound. As applications we find knot diagrammatic upper bounds on t...

We discuss bounds on the cusp volume and the length of the meridian of hyperbolic knots in terms of the topology of essential surfaces spanned by the knots. In many cases (e.g. when the knot is is ``adequate”) these bounds are obtained from knot diagrams. We will also discuss some applications to Dehn surgery.

We survey results and conjectures about topological properties and quantities of knot complements detected by the degree of colored Jones polynomials.

We establish a characterization of adequate knots in terms of the degree of their colored Jones polynomial.
We show that, assuming the Strong Slope conjecture, our characterization can be reformulated in terms of ``Jones slopes" of knots and the
essential surfaces that realize the slopes.
For alternating knots the reformulated characterization fol...

Slides from a talk on determining the geometric type of surfaces using the Jones polynomial

Slides from a talk given at the conference ``Invariants in Low dimensional Geometry" at Gazi University, Ankara, Turkey, August 2015

Slides from a a talk on conjectures about the degree of the Colored Jones polynomial and recent progress on them.

This is an introductory talk for graduate students given at the graduate student workshop before the Spring 2015 Redbud Topology conference at OkState.

This is an introductory talk given at the Graduate student workshop before the Spring 2015 Redbud Topology Conference at OKState University.

We will discuss a way to “re-package" the colored Jones polynomial knot invariants that allows to read some of the geometric properties of knot complements they detect.

We continue our study of the degree of the colored Jones polynomial under
knot cabling started in "Knot Cabling and the Degree of the Colored Jones
Polynomial" (arXiv:1501.01574). Under certain hypothesis on this degree, we
determine how the Jones slopes and the linear term behave under cabling. As an
application we verify Garoufalidis' Slope Conje...

We study the behavior of the degree of the colored Jones polynomial and the
boundary slopes of knots under the operation of cabling. We show that, under
certain hypothesis on this degree, if a knot $K$ satisfies Garoufalidis' Slope
Conjecture then a $(p, q)$-cable of $K$ satisfies the conjecture, provided that
$p/q$ is not a Jones slope of $K$. As...

We study the behavior of the degree of the colored Jones polynomial and the boundary slopes of knots under the operation of cabling. We show that, under certain hypothesis on this degree, if a knot K satisfies the Slope Conjecture then a (p, q)-cable of K satisfies the conjecture, provided that p/q is not a Jones slope of K. As an application we pr...

We provide a diagrammatic criterion for semi-adequate links to be
hyperbolic. We also give a conjectural description of the satellite
structures of semi-adequate links. One application of our result
is that the closures of sufficiently complicated positive braids are
hyperbolic links.

We give sharp two-sided linear bounds of the crosscap number (non-oriented
genus) of alternating links in terms of their Jones polynomial. Our estimates
are often exact and we use them to calculate the crosscap numbers for several
infinite families of alternating links and for several alternating knots with
up to twelve crossings. We also discuss g...

Let K' be a knot that admits no cosmetic crossing changes and let C be a
non-trivial, prime, non-cable knot. Then any knot that is a satellite of C with
winding number zero and pattern K' admits no cosmetic crossing changes. As a
consequence we prove the nugatory crossing conjecture for Whitehead doubles of
prime, non-cable knots.

The extreme degrees of the colored Jones polynomial of any link are bounded in terms of concrete data from any link diagram. It is known that these bounds are sharp for semi-adequate diagrams. One of the goals of this paper is to show the converse; if the bounds are sharp then the diagram is semi-adequate. As a result, we use colored Jones link pol...

We provide a diagrammatic criterion for semi-adequate links to be hyperbolic.
We also give a conjectural description of the satellite structures of
semi-adequate links. One application of our result is that the closures of
sufficiently complicated positive braids are hyperbolic links.

Theorem 5.14 reduces the problem of computing the Euler characteristic of the guts of M
A
to counting how many complex EPDs are required to span the I-bundle of the upper polyhedron. Our purpose in this chapter is to recognize such EPDs from the structure of the all-A state graph \({\mathbb{G}}_{A}\). The main result is Theorem 6.4, which describes...

Recall that we are trying to relate geometric and topological aspects of the knot complement \({S}^{3} \setminus K\) to quantum invariants and diagrammatic properties. So far, we have identified an essential state surface S
A
, and we have found a polyhedral decomposition of \({M}_{A}\,=\,{S}^{3}\setminus \setminus {S}_{A}\).

In this chapter, we start with a connected link diagram and explain how to construct state graphs and state surfaces. We cut the link complement in S
3 along the state surface, and then describe how to decompose the result into a collection of topological balls whose boundaries have a checkerboard coloring.

In this chapter, we study state surfaces of Montesinos links, and calculate their guts. Our main result is Theorem 8.6. In that theorem, we show that for every sufficiently complicated Montesinos link K, either K or its mirror image admits an A-adequate diagram D such that the quantity \(\vert \vert {E}_{c}\vert \vert \) of Definition 5.9 vanishes.

In this chapter, which is independent from the remaining chapters, we will restrict ourselves to A-adequate diagrams D(K) for which the polyhedral decomposition includes no non-prime arcs or switches. In this case, one can simplify the statement of Theorem 5.14 and give an easier combinatorial estimate for the guts of M
A
. This is done in Theorem...

This chapter contains one of the main results of the manuscript, namely a calculation of the Euler characteristic of the guts of M
A
in Theorem 5.14. The calculation will be in terms of the number of essential product disks (EPDs) for M
A
which are complex, as in Definition 5.2, below.

Recall that \({M}_{A} = {S}^{3}\setminus \setminus {S}_{A}\) is S
3 cut along the surface S
A
. In the last chapter, starting with a link diagram D(K), we obtained a prime decomposition of M
A
into 3-balls. One of our goals in this chapter is to show that, if D(K) is A-adequate (see Definition 1.1 on p. 4), each of these balls is a checkerboard col...

We prove that the property of admitting no cosmetic crossing changes is
preserved under the operation of inserting full twists in the strings of closed
braids and the operation of forming certain satellites of winding number zero.
As a consequences of the main results, we prove the nugatory crossing
conjecture for twisted fibered braids, for closed...

In this final chapter, we state some questions that arose from this work and speculate about future directions related to this project. In Sect. 10.1, we discuss modifications of the diagram D that preserve A-adequacy. In Sect. 10.2, we speculate about using normal surface theory in our polyhedral decomposition of M
A
to attack various open problem...

In this chapter, we will use the calculations of \(\mathrm{guts}({S}^{3}\setminus \setminus {S}_{A})\) obtained in earlier chapters to relate the geometry of A-adequate links to diagrammatic quantities and to Jones polynomials. In Sect. 9.1, we combine Theorem 5.14 with results of Agol et al. [6] to obtain bounds on the volumes of hyperbolic A-adeq...

This paper continues our study, initiated in [arXiv:1108.3370], of
essential state surfaces in link complements that satisfy a mild
diagrammatic hypothesis (homogeneously adequate). For hyperbolic links,
we show that the geometric type of these surfaces in the Thurston
trichotomy is completely determined by a simple graph--theoretic
criterion in te...

We study framed links in irreducible 3-manifolds that are $Z$-homology
3-spheres or atoroidal $Q$-homology 3-spheres. We calculate the dual of the
Kauffman skein module over the ring of two variable power series with complex
coefficients. For links in $S^3$ we give a new construction of the classical
Kauffman polynomial.

This paper is a brief overview of recent results by the authors relating
colored Jones polynomials to geometric topology. The proofs of these results
appear in the papers [arXiv:1002.0256] and [arXiv:1108.3370], while this survey
focuses on the main ideas and examples.

This monograph derives direct and concrete relations between colored Jones
polynomials and the topology of incompressible spanning surfaces in knot and
link complements. Under mild diagrammatic hypotheses that arise naturally in
the study of knot polynomial invariants (A- or B-adequacy), we prove that the
growth of the degree of the colored Jones p...

We study cosmetic crossings in knots of genus one and obtain obstructions to
such crossings in terms of knot invariants determined by Seifert matrices. In
particular, we prove that for genus one knots the Alexander polynomial and the
homology of the double cover branching over the knot provide obstructions to
cosmetic crossings. As an application w...

We show that for genus one knots the Alexander polynomial and the homology of
the double cover branching over the knot provide obstructions to cosmetic
crossings. As an application we prove the nugatory crossing conjecture for the
negatively twisted, positive Whitehead doubles of all knots. We also verify the
conjecture for several families of pret...

Garoufalidis conjectured a relation between the boundary slopes of a knot and
its colored Jones polynomials. According to the conjecture, certain boundary
slopes are detected by the sequence of degrees of the colored Jones
polynomials. We verify this conjecture for adequate knots, a class that vastly
generalizes that of alternating knots.

In recent years, several families of hyperbolic knots have been shown to have both volume and $\lambda_1$ (first eigenvalue of the Laplacian) bounded in terms of the twist number of a diagram, while other families of knots have volume bounded by a generalized twist number. We show that for general knots, neither the twist number nor the generalized...

This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured-torus bundles, 4-punctured sphere
bundles, and two-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey
tessellation of the hyperbolic plane. The bounds on cusp area lead to explicit bounds on the volume o...

We obtain bounds on hyperbolic volume for periodic links and Conway sums of alternating tangles. For links that are Conway sums we also bound the hyperbolic volume in terms of the coefficients of the Jones polynomial.

The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobás–Riordan–Tutte polynomial generalizes the Tutte polynomial of graphs to graphs that are embedded in closed oriented surfaces of higher genus.In this paper...

A classical result states that the determinant of an alternating link is equal to the number of spanning trees in a checkerboard graph of an alternating connected projection of the link. We generalize this result to show that the determinant is the alternating sum of the number of quasi-trees of genus j of the dessin of a non-alternating link.
Furt...

We investigate the conjectural relations between the Reshetikhin-Turaev-Witten quantum SU(2) invariants and the volume of hyperbolic 3-manifolds. Given a finite set of sufficiently large positive integers, say J, we construct examples of closed hyperbolic 3-manifolds with the same invariants at all levels in J and different volume.

We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group.