arXiv subjects
Publications and source records attributed to Efstratia Kalfagianni.
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 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 the Figure-eight knot and the Borromean rings complement determine the corresponding hyperbolic volumes. Our calculations also exhibit new phenomena of asymptotic behavior of values of the colored Jones polynomials that seem not to be predicted by neither the Kashaev-Murakami-Murakami volume conjecture and various of its generalizations nor by Zagier's quantum modularity conjecture. We conjecture that the asymptotics of the Turaev-Viro invariants of any link complement determine the simplicial volume of the link, and verify it for all knots with zero simplicial volume. Finally we observe that our simplicial volume conjecture is stable under connect sum and split unions of links.
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 geometric structures of link complements. State surfaces also provide a tool for studying relations between Jones polynomials and topological invariants, such as the crosscap number or invariants coming from geometric structures on link complements (e.g. hyperbolic volume). This article is brief survey on some of the recent applications of state surfaces.
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 that satisfy the Strong Slope Conjecture (Jones surfaces). We also present numerical and experimental evidence supporting a stronger such relation which we state as an open question.
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 area. We give sample applications and state some open questions and conjectures.
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 the meridian length and the cusp volume of hyperbolic adequate knots and we obtain new large families of knots with meridian lengths bounded above by four. We also discuss applications of our results to Dehn surgery.
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.
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 follows by recent work of J. Greene and J. Howie.
We give sharp two-sided linear bounds of the crosscap number (non-orientable 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 generalizations of our results for classes of non-alternating links.
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 prove the Slope Conjecture for iterated cables of adequate knots and for iterated torus knots. Furthermore we show that, for these knots, the degree of the colored Jones polynomial also determines the topology of a surface that satisfies the Slope Conjecture. We also state a conjecture suggesting a topological interpretation of the linear terms of the degree of the colored Jones polynomial (Conjecture \ref{conj}), and we prove it for the following classes of knots:iterated torus knots and iterated cables of adequate knots, iterated cables of several non-alternating knots with up to nine crossings, pretzel knots of type $(-2, 3, p)$ and their cables, and two-fusion knots.
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 Conjecture and a conjecture of the authors for cables of a two-parameter family of closed 3-braids called 2-fusion knots.
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.
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 polynomials to extract an invariant that detects semi-adequate links and discuss some applications.
We prove that the property of admitting no cosmetic crossing changes is preserved under the operation of forming certain satellites of winding number zero. We also define strongly cosmetic crossing changes and we discuss their behavior under the operation of inserting full twists in the strings of closed braids.
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 terms of a certain spine of the surfaces. For links with A- or B-adequate diagrams, the geometric type of the surface is also completely determined by a coefficient of the colored Jones polynomial of the link.
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 polynomials is a boundary slope of an essential surface in the knot complement. We show that certain coefficients of the polynomial measure how far this surface is from being a fiber in the knot complement; in particular, the surface is a fiber if and only if a particular coefficient vanishes. Our results also yield concrete relations between hyperbolic geometry and colored Jones polynomials: for certain families of links, coefficients of the polynomials determine the hyperbolic volume to within a factor of 4. Our approach is to generalize the checkerboard decompositions of alternating knots. Under mild diagrammatic hypotheses (A- or B-adequacy), we show that the checkerboard knot surfaces are incompressible, and obtain an ideal polyhedral decomposition of their complement. We employ normal surface theory to establish a dictionary between the pieces of the JSJ decomposition of the surface complement and the combinatorial structure of certain spines of the checkerboard surface (state graphs). Since state graphs have previously appeared in the study of Jones polynomials, our setting and methods create a bridge between quantum and geometric knot invariants.
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.
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 we prove the nugatory crossing conjecture for twisted Whitehead doubles of non-cable knots. We also verify the conjecture for several families of pretzel knots and all genus one knots with up to 12 crossings.