arXiv subjects

Efstratia Kalfagianni

Publications and source records attributed to Efstratia Kalfagianni.

At least 19 recordsLinked to original sources

arXiv2026

Seifert cobordisms and the Chen-Yang volume conjecture

We study the large $r$ asymptotic behavior of the Turaev-Viro invariants $TV_r(M; e^{\frac{2πi}{r}})$ of 3-manifolds with toroidal boundary, under the operation of gluing a Seifert-fibered 3-manifold along a component of $\partial M$. We show that the Turaev-Viro invariants volume conjecture is closed under this operation. As an application we prove the volume conjecture for all Seifert fibered 3-manifolds with boundary and for large classes of graph 3-manifolds.

math.GT↗

arXiv2026

Ivanov's finite-index Frattini conjecture for subgroups of mapping class groups

Let $S_g$ be a closed oriented surface of genus $g$. Ivanov conjectured that if $G$ a finitely generated subgroup of $\mathrm{Mod}(S_g)$ its finite-index Frattini subgroup $Φ_f(G)$ is nilpotent. We prove this conjecture, using recent partial results of the first author about the Andersen-Masbaum-Ueno conjecture.

math.GT↗

arXiv2026

Torsion in skein modules and shared knot surgeries

For every $g>0$, we construct a closed Haken 3-manifold $M_g$, whose only connected, incompressible surface is a non-separating surface of genus $g$ and discuss the torsion behavior of the Kauffman bracket skein module $§(M_g,\Z[A^{\pm 1}])$. We also use properties of shared surgeries between knots to estimate (and often compute) the dimension of the Kauffman bracket skein module over $\Q(A)$, for families of hyperbolic 3-manifolds obtained by surgery on double twist knots. We have similar computations for some surgeries on several knot complements in the SnapPy census of cusped hyperbolic manifolds with triangulations up to 9 tetrahedra.

math.GT↗

arXiv2026

On the quantum $\mathfrak{sl}_3$ invariant of positive links

We use the skein theory of $\mathfrak{sl}_3$-webs to study the properties of the quantum $\mathfrak{sl}_3$-link polynomial of positive links. We give explicit formulae for the three leading terms of the polynomial on positive links in terms of diagrammatic quantities of their positive diagrams. We show that a positive link is fibered if and only the second coefficient of the polynomial is equal to one. We also show that the third coefficient provides obstructions to representing links by positive braids.

math.GT↗

arXiv2025

Pants complex, TQFT and hyperbolic geometry

We introduce a coarse perspective on relations of the $SU(2)$-Witten-Reshetikhin-Turaev TQFT, the Weil-Petersson geometry of the Teichmüller space, and volumes of hyperbolic 3-manifolds. Using data from the asymptotic expansions of the curve operators in the skein theoretic version of the $SU(2)$-TQFT, we define the quantum intersection number between pants decompositions of a closed surface. We show that the quantum intersection number admits two sided bounds in terms of the geometric intersection number and we use it to obtain a metric on the pants graph of surfaces. Using work of Brock we show that the pants graph equipped with this metric is quasi-isometric to the Teichmüller space with the Weil-Petersson metric and that the translation length of our metric provides two sided linear bounds on the volume of hyperbolic fibered manifolds. We briefly discuss how these relations are interpeted from the view point of $SU(2)$-character varieties of 3-manifolds. We also obtain a characterization of pseudo-Anosov mapping classes in terms of asymptotics of the quantum intersection number under iteration in the mapping class group and relate these asymptotics with stretch factors. We also discuss how these results fit with a conjecture of Andersen, Masbaum and Ueno about quantum representations of mapping class groups.

math.GT↗

arXiv2025

Skein modules and character varieties of Seifert manifolds

We show that the Kauffman bracket skein module of a closed Seifert fibered 3-manifold $M$ is finitely generated over $\mathbb Z[A^{\pm 1}]$ if and only if $M$ is irreducible and non-Haken. We analyze in detail the character varieties $X(M)$ of such manifolds and show that under mild conditions they are reduced. We compute the Kauffman bracket skein modules for these $3$-manifolds (over $\mathbb Q(A)$) and show that their dimensions coincide with $|X(M)|.$

math.GT↗

arXiv2025

Kauffman bracket skein modules of small 3-manifolds

The proof of Witten's finiteness conjecture established that the Kauffman bracket skein modules of closed $3$-manifolds are finitely generated over $\mathbb Q(A)$. In this paper, we develop a novel method for computing these skein modules. We show that if the skein module $S(M,\mathbb Q[A^{\pm 1}])$ of $M$ is tame (e.g. finitely generated over $\mathbb Q[A^{\pm 1}]$), and the $SL(2,\mathbb C)$-character variety is reduced, then the dimension $\dim_{\mathbb Q(A)}\, S(M, \mathbb Q(A))$ is the number of closed points in this character variety. This, in particular, verifies a conjecture in the literature that relates the dimension $\dim_{\mathbb Q(A)}\, S(M, \mathbb Q(A))$ to the Abouzaid-Manolescu $SL(2,\mathbb C)$-Floer theoretic invariants, for large families of 3-manifolds. We also prove a criterion for reduceness of character varieties of closed $3$-manifolds and use it to compute the skein modules of Dehn fillings of $(2,2n+1)$-torus knots and of the figure-eight knot. The later family gives the first instance of computations of skein modules for closed hyperbolic 3-manifolds. We also prove that the skein modules of rational homology spheres have dimension at least $1$ over $\mathbb Q(A)$.

math.GT↗

arXiv2024

Crossing numbers of cable knots

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 also determine the crossing number of the connected sum of any adequate knot with a $2$-cable of an adequate knot.

math.GT↗

arXiv2024

Constructions of $q$-hyperbolic 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-Anosov mapping classes acting on surfaces of any genus and with one boundary component that satisfy the conjecture.

math.GT↗

arXiv2023

Jones diameter and crossing number of knots

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 knots. More specifically, we exhibit minimal crossing number diagrams for untwisted Whitehead doubles of zero-writhe adequate knots. This allows us to determine the crossing number of untwisted Whitehead doubles of any amphicheiral adequate knot, including, for instance, the Whitehead doubles of the connected sum of any alternating knot with its mirror image. We also determine the crossing number of the connected sum of any adequate knot with an untwisted Whitehead double of a zero-writhe adequate knot.

math.GT↗

arXiv2023

Volumes of fibered 2-fold branched covers of 3-manifolds

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.

math.GT↗

arXiv2022

Alternating links on surfaces and volume bounds

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 Lackenby. Howie and Purcell showed that a similar lower bound holds for weakly generalised alternating knots. In this paper, we show that a generalisation of the upper volume bound does not hold, by producing a family of weakly generalised alternating knots in the 3-sphere with fixed twist number but unbounded volumes. As a corollary, generalised alternating knots can have arbitrarily small cusp density, in contrast with usual alternating knots whose cusp densities are bounded away from zero due to Lackenby and Purcell. On the other hand, we show that the twist number of a weakly generalised alternating projection does gives two sided linear bounds on volume inside a thickened surface; we state some related open questions.

math.GT↗

arXiv2021

Guts, volume and Skein Modules of 3-manifolds

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 surface, this Kauffman bracket function leads to a Jones-type polynomial that is an isotopy invariant of links. We show that coefficients of this polynomial provide 2-sided linear bounds on the volume of hyperbolic alternating links in the thickened surface. As a corollary of the proof of this result, we deduce that the twist number of a reduced, twist reduced, checkerboard alternating link projection with disk regions, is an invariant of the link.

math.GT↗

arXiv2020

Cosets of monodromies and quantum representations

We use geometric methods to show that given any $3$-manifold $M$, and $g$ a sufficiently large integer, the mapping class group $\mathrm{Mod}(Σ_{g,1})$ contains a coset of an abelian subgroup of rank $\lfloor \frac{g}{2}\rfloor,$ consisting of pseudo-Anosov monodromies of open-book decompositions in $M.$ We prove a similar result for rank two free cosets of $\mathrm{Mod}(Σ_{g,1}).$ These results have applications to 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 elements are fibered 3-manifolds that satisfy a weak form of the Turaev-Viro invariants volume conjecture.

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arXiv2020

Remarks on Jones slopes and surfaces of knots

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.

math.GT↗

arXiv2020

Growth of quantum 6j-symbols and applications to the Volume Conjecture

We prove the Turaev-Viro invariants volume conjecture for a "universal" class of cusped hyperbolic 3-manifolds that produces all 3-manifolds with empty or toroidal boundary by Dehn filling. This leads to two-sided bounds on the volume of any hyperbolic 3-manifold with empty or toroidal boundary in terms of the growth rate of the Turaev-Viro invariants of the complement of an appropriate link contained in the manifold. We also provide evidence for a conjecture of Andersen, Masbaum and Ueno (AMU conjecture) about certain quantum representations of surface mapping class groups. A key step in our proofs is finding a sharp upper bound on the growth rate of the quantum $6j-$symbol evaluated at $q=e^{\frac{2πi}{r}}.$

math.GT↗

arXiv2019

Quantum representations and monodromies of fibered links

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. We show that if the $r$-growth of $|TV_r(M)|$ for a hyperbolic 3-manifold $M$ that fibers over the circle is exponential, then the monodromy of the fibration of $M$ satisfies the AMU conjecture. Building on earlier work \cite{DK} we give broad constructions of (oriented) hyperbolic fibered links, of arbitrarily high genus, whose $SO(3)$-Turaev-Viro invariants have exponential $r$-growth. As a result, for any $g>n\geqslant 2$, we obtain infinite families of non-conjugate pseudo-Anosov mapping classes, acting on surfaces of genus $g$ and $n$ boundary components, that satisfy the AMU conjecture. We also discuss integrality properties of the traces of quantum representations and we answer a question of Chen and Yang about Turaev-Viro invariants of torus links.

math.GT↗

arXiv2019

Gromov norm and Turaev-Viro invariants of 3-manifolds

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 the Gromov norm of $M$. The proof combines TQFT techniques, geometric decomposition theory of 3-manifolds and analytical estimates of $6j$-symbols. We obtain topological criteria that can be used to check whether the growth is actually exponential; that is one has $\log| TV_r (M)|\geqslant B \ r$, for some $B>0$. We use these criteria to construct infinite families of hyperbolic 3-manifolds whose $SO(3)$ Turaev-Viro invariants grow exponentially. These constructions are essential for the results of [DK:AMU] where the authors make progress on a conjecture of Andersen, Masbaum and Ueno about the geometric properties of surface mapping class groups detected by the quantum representations. We also study the behavior of the Turaev-Viro invariants under cutting and gluing of 3-manifolds along tori. In particular, we show that, like the Gromov norm, the values of the invariants do not increase under Dehn filling and we give applications of this result on the question of the extent to which relations between the invariants $TV_r$ and hyperbolic volume are preserved under Dehn filling. Finally we give constructions of 3-manifolds, both with zero and non-zero Gromov norm, for which the Turaev-Viro invariants determine the Gromov norm.

math.GT↗