arXiv research

The study calculates the average genus of rational knots and links.

New model shows average genus of 2-bridge knots grows linearly with crossing number.

The study calculates the average genus of 2-bridge knots based on their crossing numbers.

Average signature of 2-bridge knots approximates sqrt(2c/π).

Lower bounds on average genus of 2-bridge knots found.

The study calculates average crosscap numbers for 2-bridge knots.

We prove that the expected value of the ratio between the smooth four-genus and the Seifert genus of two-bridge knots tends to zero as the crossing number tends to infinity.

Estimates scalar curvature on moduli space as genus grows.

GOE statistics emerge from surface moduli space averages.

The paper shows Gaussian fluctuations in eigenvalue statistics of random hyperbolic surfaces.

The study examines averages of Laplacian determinants over large genus moduli spaces.

In the planar limit of the 't Hooft expansion, the Wilson-loop average in 3d Chern-Simons theory (i.e. the HOMFLY polynomial) depends in a very simple way on representation (the Young diagram), so that the (knot-dependent) Ooguri-Vafa partition function becomes a trivial KP tau-function. We study higher genus correctio…

Sparse curves on surfaces grow at a specific intermediate rate.

Study on 3-manifolds with nonnegative scalar curvature and positive harmonic functions.

New knots found with Seifert genus not matching minimal genus Seifert surfaces.

This paper analyzes the distribution of genera in 2-bridge knots and proves their asymptotic normality.

Construct divide knots with specific genus properties.

The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of K coming from the knot Floer complex of K. As an application, we prove that ther…

The paper generalizes the $T$-genus to characterize slice knots and slice genus.

Characterizes the Z-genus of boundary links using Blanchfield forms.

The concordance genus of a knot is the least genus of any knot in its concordance class. Although difficult to compute, it is a useful invariant that highlights the distinction between the three-genus and four-genus. In this paper we define and discuss the stable concordance genus of a knot, which describes the behavio…

Exact asymptotic value of Weil-Petersson volumes computed for large genus surfaces.

Classifies 3-braid knots with maximal 4-genus using McCoy's method.

Study shows non-equivariant and equivariant non-orientable 4-genus of periodic knots can differ.

Study Khovanov homology of Turaev genus one links, finding a trivial summand.

Contact connected sums do not increase support genus.

The concordance genus of a knot is the least genus of any knot in its concordance class. It is bounded above by the genus of the knot, and bounded below by the slice genus, two well-studied invariants. In this paper we consider the concordance genus of 11--crossing prime knots. This analysis resolves the concordance ge…

This article is about the graph genus of certain well studied graphs in surface theory: the curve, pants and flip graphs. We study both the genus of these graphs and the genus of their quotients by the mapping class group. The full graphs, except for in some low complexity cases, all have infinite genus. The curve grap…

Classifies knot traces with specific trisection genus limits.

Classifies Teichmüller curves in genus three from genus two.

The paper proves trisection genus of Akbulut cork and constructs many corks with trisection genus 3.

New expanders for mean curvature flow contradict genus-reduction conjecture.

Study shows systole behavior changes significantly for large genus hyperbolic surfaces.

Lower bounds on rational slice genus using Heegaard Floer invariants.

Survey of minimal genus problem progress.

Algorithm calculates genus of embedded graphs on surfaces.

In this paper, we develop a lower bound for the double slice genus of a knot using Casson-Gordon invariants. As an application, we show that the double slice genus can be arbitrarily larger than twice the slice genus. As an analogue to the double slice genus, we also define the superslice genus of a knot, and give both…

For each g > 2 and h > 1, we explicitly construct (1) fiber sum indecomposable relatively minimal genus g Lefschetz fibrations over genus h surfaces whose monodromies lie in the Torelli group, (2) fiber sum indecomposable genus g surface bundles over genus h surfaces whose monodromies are in the Torelli group (provided…

The paper explores the genus of surfaces in complex projective spaces and improves minimal genus bounds.

Shows large unknotting number for simple knots.

We introduce a new link invariant called the algebraic genus, which gives an upper bound for the topological slice genus of links. In fact, the algebraic genus is an upper bound for another version of the slice genus proposed here: the minimal genus of a surface in the four-ball whose complement has infinite cyclic fun…

Correspondence found between Askey-Wilson polynomials and genus-two handlebody skein module.

The paper calculates genus bounds for multibranched surfaces.

New lower bound for doubly slice genus using knot signatures.

In this paper we write explicitly the open book decompositions of links of quotient surface singularities supporting the corresponding unique Milnor fillable contact structure. The page-genus of these Milnor open books are minimal among all Milnor open books supporting the same contact structure. We also investigate wh…

An open book decomposition of a 3-manifold $M$ induces a Heegaard splitting for $M$, and the minimal genus among all Heegaard splittings induced by open book decompositions is called the \emph{open book genus} of $M$. It is conjectured by Ozbagci \cite{O} that the open book genus is additive under the connected sum of …

It is well-known that Heegaard genus is additive under connected sum of 3-manifolds. We show that Heegaard genus of contact 3-manifolds is not necessarily additive under contact connected sum. We also prove some basic properties of the contact genus (a.k.a. open book genus) of 3-manifolds, and compute this invariant fo…

Wilson-loop averages in Chern-Simons theory (HOMFLY polynomials) can be evaluated in different ways -- the most difficult, but most interesting of them is the hypercube calculus, the only one applicable to virtual knots and used also for categorification (higher-dimensional extension) of the theory. We continue the stu…