arXiv research

Lower bounds on rational slice genus using Heegaard Floer invariants.

The paper reconfirms a lower bound on rational genus using Heegaard Floer homology.

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

If K is a rationally null-homologous knot in a 3-manifold M, the rational genus of K is the infimum of -χ(S)/2p over all embedded orientable surfaces S in the complement of K whose boundary wraps p times around K for some p (hereafter: S is a p-Seifert surface for K). Knots with very small rational genus can be constru…

The genus of satellite tunnel number one knots and torti-rational knots is computed using the tools introduced by Floyd and Hatcher. An implementation of an algorithm is given to compute genus and slopes of minimal genus Seifert surfaces for such knots.

Using the mapping cone of a rational surgery, we give several obstructions for Seifert fibered surgeries, including obstructions on the Alexander polynomial, the knot Floer homology, the surgery coefficient and the Seifert and four-ball genus of the knot.

A torti-rational knot, denoted by K(2a,b|r), is a knot obtained from the 2-bridge link B(2a,b) by applying Dehn twists an arbitrary number of times, r, along one component of B(2a,b). We determine the genus of K(2a,b|r) and solve a question of when K(2a,b|r) is fibred. In most cases, the Alexander polynomials determine…

Rationally null-homologous links in Seifert fibered spaces may be represented combinatorially via labeled diagrams. We introduce an additional condition on a labeled link diagram and prove that it is equivalent to the existence of a rational Seifert surface for the link. In the case when this condition is satisfied, we…

Method to create rational Seifert surfaces for knots in Lens space.

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

Study on invariant Seifert surfaces for strongly invertible knots, showing large gaps in genus.

Study shows Seifert fibered spaces don't bound rational homology balls.

The paper shows knots with specific properties have smaller 4-genus.

Characterizes symplectic rational homology ball fillings of Seifert fibered spaces.

Proves prime theta-curves for knots on minimal genus surfaces.

The paper characterizes symplectic fillings of Seifert 3-manifolds using rational blowdowns.

Roberts proved that a family of alternating, arborescent, prime knots each have at least $2^{2n-1}$ distinct minimal genus Seifert surfaces, where $n$ is the genus of the knot in question. We give a subfamily of these knots that have exactly this many minimal genus Seifert surfaces.

Legendrian surgery describes canonical contact structures and calculates Gompf's θ-invariant.

The study sets constraints on 4-manifold forms linked to specific invariants.

A Seifert surface F for a knot K is free if the complement of F is a handlebody (i.e., has free fundamental group). The free genus of K is the minimum genus among all free Seifert surfaces for K. In this paper we show that there exist families of hyperbolic knots with arbitrarily large volume, which each have free genu…

Under a simple assumption on Seifert surfaces, we characterise knots whose stable topological 4-genus coincides with the genus.

We verify that the rational blow-down schemes along certain Seifert fibered 3-manifolds found by the second author, Szabo and Wahl are, in fact, symplectic operations.

Same-genus Seifert surfaces for non-split alternating links are smoothly isotopic.

Establishes a rank inequality between knot Floer homologies of freely 2-periodic knots and their quotients.

In this note we study the Seifert rational homology spheres with two complementary legs, i.e. with a pair of invariants whose fractions add up to one. We give a complete classification of the Seifert manifolds with 3 exceptional fibers and two complementary legs which bound rational homology balls. The result translate…

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

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 introduces new invariants for genus one knots and surfaces.

This paper classifies curves in genus two handlebodies.

Let W -> X be a real smooth projective 3-fold fibred by rational curves. J. Kollár proved that, if W(R) is orientable, then a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces. Our Main Theorem, answering in the affirmative three questions of Kollár, gives s…

New lower bound for knot genus using Links-Gould invariant.

Researchers calculate the volume of Seifert representations for graph manifolds and their covers.

Let K be a knot in S^3 of genus g and let n>0. We show that if rk HFK(K,g) < 2^{n+1} (where HFK denotes knot Floer homology), in particular if K is an alternating knot such that the leading coefficient a_g of its Alexander polynomial satisfies |a_g| <2^{n+1}, then K has at most n pairwise disjoint non-isotopic genus g …

To a Seifert matrix of a knot K one can associate a matrix w(K) with entries in the rational function field, Q(t). The Murasugi, Milnor, and Levine-Tristram knot signatures, all of which provide bounds on the 4-genus of a knot, are determined by w(K). More generally, the minimal rank of a representative of the class re…

Simple proof of knot genus theorem using Alexander polynomial.

We write a formula for the LMO invariant of a rational homology sphere presented as a rational surgery on a link in S^3. Our main tool is a careful use of the Aarhus integral and the (now proven) "Wheels" and "Wheeling" conjectures of B-N, Garoufalidis, Rozansky and Thurston. As steps, side benefits and asides we give …

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact structures for every r not= 2g_s(K)-1, where g_s(K) is the slice genus…

We calculate the RT-invariants of all oriented Seifert manifolds directly from surgery presentations. We work in the general framework of an arbitrary modular category as in [V. G. Turaev, Quantum invariants of knots and 3--manifolds, de Gruyter Stud. Math. 18, Walter de Gruyter (1994)], and the invariants are expresse…

The article proves properties of Seifert links and their cyclic branched covers.

Homogeneous links were introduced by Peter Cromwell, who proved that the projection surface of these links, that given by the Seifert algorithm, has minimal genus. Here we provide a different proof, with a geometric rather than combinatorial flavor. To do this, we first show a direct relation between the Seifert matrix…

This paper presents a new algorithm "A" for constructing Seifert surfaces from n-bridge projections of links. The algorithm produces minimal complexity surfaces for large classes of braids and alternating links. In addition, we consider a family of knots for which the canonical genus is strictly greater than the genus,…

Instanton Floer homology matches Heegaard Floer for almost-rational plumbings.

A Seifert surface for a knot K is called canonical if it can be built by applying Seifert's algorithm to some projection of K. The canonical genus of K is the smallest genus of a surface so obtained. In this paper we show that there is a bound on the volume of a hyperbolic knot which admits a canonical surface of genus…

It is known that every closed oriented 3-manifold is homology cobordant to a hyperbolic 3-manifold. By contrast we show that many homology cobordism classes contain no Seifert fibered 3-manifold. This is accomplished by determining the isomorphism type of the rational cohomology ring of all Seifert fibered 3-manifolds …

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

Authors show that genus defects of Hopf arborescent links are decidable.

We describe a procedure for creating infinite families of hyperbolic knots having unique minimal genus Seifert surface. A large subset of these knots have the further property that the surface cannot be the sole compact leaf of a depth one foliation of the knot exterior.

A virtual knot that has a homologically trivial representative $\mathscr{K}$ in a thickened surface $Σ\times [0,1]$ is said to be an almost classical (AC) knot. $\mathscr{K}$ then bounds a Seifert surface $F\subset Σ\times [0,1]$. Seifert surfaces of AC knots are useful for computing concordance invariants and slice ob…