arXiv research

Characterizes values of slice-torus invariants related to knot genus.

Squeezed knots are slices of minimal cobordisms; obstructions come from quantum knot invariants.

Defines super stable maps and proves quotient superorbifolds for genus zero.

This paper contains the results of efforts to determine values of the smooth and the topological slice genus of 11- and 12-crossing knots. Upper bounds for these genera were produced by using a computer to search for genus one concordances between knots. For the topological slice genus further upper bounds were produce…

An important difference between high dimensional smooth manifolds and smooth 4-manifolds that in a 4-manifold it is not always possible to represent every middle dimensional homology class with a smoothly embedded sphere. This is true even among the simplest 4-manifolds: $X_0(K)$ obtained by attaching an $0$-framed 2-h…

This paper presents evidence supporting the surprising conjecture that in the topological category the slice genus of a satellite knot $P(K)$ is bounded above by the sum of the slice genera of $K$ and $P(U)$. Our main result establishes this conjecture for a variant of the topological slice genus, the $\mathbb{Z}$-slic…

New examples show algebraically slice knots with specific genus bounds.

New knots found that are 4-genus minimal.

Study knots in definite 4-manifolds using minimum-genus bounds.

New homomorphism from Khovanov homology gives slice genus bounds.

Study framed links in simply connected 4-manifolds, linking to exotic phenomena.

We construct an infinite family of topologically slice 2--component boundary links $\ell_i$, none of which is smoothly concordant to a split link, such that $g_4(\ell_i)=i$.

Characterizes fractional Dehn twist coefficient and proves slice-Bennequin inequality.

The paper bounds the genus of surfaces in four-manifolds with indefinite forms.

By considering negative surgeries on a knot $K$ in $S^3$, we derive a lower bound to the non-orientable slice genus $γ_4(K)$ in terms of the signature $σ(K)$ and the concordance invariants $V_i(\overline{K})$, which strengthens a previous bound given by Batson, and which coincides with Ozsváth-Stipsicz-Szabó's bound in…

We define a filtration of the smooth concordance group based on the genus of representative knots. We use the Heegaard Floer epsilon and Upsilon invariants to prove the quotient groups with respect to this filtration are infinitely generated. Results are applied to three infinite families of topologically slice knots.

New knot concordance invariants from Seiberg-Witten theory bound slice genus.

The paper defines new knot genera and finds bounds for stabilization distances.

Smooth torus actions on moduli spaces of super stable curves and maps of genus zero.

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

New lower bound for doubly slice genus using knot signatures.

We classify the positive definite intersection forms that arise from smooth 4-manifolds with torsion-free homology bounded by positive integer surgeries on the right-handed trefoil. A similar, slightly less complete classification is given for the (2,5)-torus knot, and analogous results are obtained for integer surgeri…

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…

Lower bounds on rational slice genus using Heegaard Floer invariants.

Sharp knots and iterated cables lead to ribbon knots or failure of slice-ribbon conjecture.

New invariants improve Heegaard Floer slice genus and clasp number bounds.

In this paper, we compute the slice genus for many low-crossing virtual knots. For instance, we show that 1295 out of 92800 virtual knots with 6 or fewer crossings are slice, and that all but 248 of the rest are not slice. Key to these results are computations of Turaev's graded genus, which we show extends to give an …

In a previous paper we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these cohomology operations commute with cobordism maps on Khovanov homology. As a consequence we obtain a refinement of Rasmussen's …

We study the double slice genus of a knot, a natural generalization of slice genus. We define a notion called band number, a natural generalization of band unknotting number, and prove it is an upper bound on double slice genus. Our bound is based on an analysis of broken surface diagrams and embedding properties of 3-…

Invariants from surface Khovanov-Jacobsson classes help detect knots and slices.

Local knots can't bound smaller surfaces in rational homology 3-spheres.

The stable 4-genus of a knot K in 3-space is the limiting value of g_4(nK)/n, where g_4 denotes the 4-genus and n goes to infinity. This induces a seminorm on CQ, the concordance group tensored with the rational numbers. Basic properties of the stable genus are developed, as are examples focused on understanding the un…

Study shows knots can have large genus difference from concordance.

New bounds on slice genus from knot invariants.

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…

We show that perturbing the definition of sl(n) Khovanov-Rozansky link homology gives a lower bound on the slice genus of a knot. As a corollary this yields another proof of Milnor's conjecture on the slice genus of torus knots.

We define Casson-Gordon sigma-invariants for links and give a lower bound of the slice genus of a link in terms of these invariants. We study as an example a family of two component links of genus h and show that their slice genus is h, whereas the Murasugi-Tristram inequality does not obstruct this link from bounding …

Study on knots, genera, and algebraic concordance groups.

The algebraic genus of a knot is an invariant that arises when one considers upper bounds for the topological slice genus coming from Freedman's theorem that Alexander polynomial one knots are topologically slice. This paper develops null-homologous twisting operations as a tool for studying the algebraic genus and, co…

Identifies doubly slice genera for 2909 prime knots with up to 12 crossings.

We show that the moduli space of genus zero stable maps is a real projective variety if the target space is a smooth convex real projective variety. We show that evaluation maps, forgetful maps are real morphisms. We analyze the real part of the moduli space.

Obstructs Legendrian knots from being slices of concordances using doubly slice genus.

New invariants refine link homology, showing large genus differences.

Study knot Floer homology to create concordance invariants and slice genus bounds.

The twisting number of a ribbon knot is at least as large as its doubly slice genus.

New invariant measures doubly slice links, disproving previous bounds.

We prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus. As an application, we derive the best possible linear estimate of the topological slice genus for torus knots with non-maximal signature invariant.

Study equivariant 4-genus of knots in symmetric 4-manifolds.