Slice Genus Research Articles

A refinement of Rasmussen’s S-invariant

In a previous work, 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 slice genus bound s for each stable cohomology operation. We show that in the case of the Steenrod square Sq2 our refinement is strictly stronger than s.

Nonorientable slice genus can be arbitrarily large

Nonorientable slice genus can be arbitrarily large

THE CONCORDANCE GENUS OF 11-CROSSING KNOTS

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 genus of 533 of the 552 prime 11-crossing knots. The appendix to the paper gives concordance diagrams for 59 knots found to be concordant to knots of lower genus, including null-concordances for the 30 11-crossing knots known to be slice.

ON THE QUANTUM FILTRATION OF THE UNIVERSAL sl(2) FOAM COHOMOLOGY

We investigate the filtered theory corresponding to the universal sl(2) foam cohomology [Formula: see text] for links, where a, h ∈ ℂ. We show that there is a spectral sequence converging to [Formula: see text] which is invariant under the Reidemeister moves, and whose E1 term is isomorphic to Khovanov homology. This spectral sequence can be used to obtain from the foam perspective an analogue of the Rasmussen invariant and a lower bound for the slice genus of a knot.

Computable bounds for Rasmussen’s concordance invariant

AbstractGiven a diagram D of a knot K, we give easily computable bounds for Rasmussen’s concordance invariant s(K). The bounds are not independent of the diagram D chosen, but we show that for diagrams satisfying a given condition the bounds are tight. As a corollary we improve on previously known Bennequin-type bounds on the slice genus.

Asymptotic concordance invariants for ergodic vector fields

We study the asymptotics of a family of link invariants on the orbits of a smooth volume-preserving ergodic vector field on a compact domain of the 3-space. These invariants, called linear saddle invariants, include many concordance invariants and generate an infinite-dimensional vector space of link invariants. In contrast, the vector space of asymptotic linear saddle invariants is 1-dimensional, generated by the asymptotic signature. We also determine the asymptotic slice genus and relate it to the asymptotic signature.

Khovanov homology and the slice genus

We use Lee’s work on the Khovanov homology to define a knot invariant s. We show that s(K) is a concordance invariant and that it provides a lower bound for the smooth slice genus of K. As a corollary, we give a purely combinatorial proof of the Milnor conjecture.

Higher-order Analogues of the Slice Genus of a Knot

For certain classes of knots we define geometric invariants called higher-order genera. Each of these invariants is a refinement of the slice genus of a knot. We find lower bounds for the higher-order genera in terms of certain von Neumann $\rho$-invariants, which we call higher-order signatures. The higher-order genera offer a refinement of the Grope filtration of the knot concordance group.

On slicing invariants of knots

The slicing number of a knot, u_s(K), is the minimum number of crossing changes required to convert K to a slice knot. This invariant is bounded above by the unknotting number and below by the slice genus g_s(K). We show that for many knots, previous bounds on the unknotting number obtained by Ozsvath and Szabo and by the author in fact give bounds on the slicing number. Livingston defined another invariant U_s(K), which takes into account signs of crossings changed to get a slice knot and which is bounded above by the slicing number and below by the slice genus. We exhibit an infinite family of knots K_n with slice genus n and Livingston invariant greater than n. Our bounds are based on restrictions (using Donaldson's diagonalisation theorem or Heegaard Floer homology) on the intersection forms of four-manifolds bounded by the double branched cover of a knot.

Unknotting sequences for torus knots

AbstractThe unknotting number of a knot is bounded from below by its slice genus. It is a well-known fact that the genera and unknotting numbers of torus knots coincide. In this paper we characterize quasipositive knots for which the genus bound is sharp: the slice genus of a quasipositive knot equals its unknotting number, if and only if the given knot appears in an unknotting sequence of a torus knot.

A slice genus lower bound from [formula omitted] Khovanov–Rozansky homology

We show that a generic perturbation of the doubly-graded Khovanov–Rozansky knot homology gives rise to a lower-bound on the slice genus of a knot. We prove a theorem about obtainable presentations of surfaces embedded in 4-space, which we use to simplify significantly our algebraic computations.

Topological minimal genus andL2–signatures

We obtain new lower bounds for the minimal genus of a locally flat surface representing a 2‐dimensional homology class in a topological 4‐manifold with boundary, using the von Neumann‐Cheeger‐Gromov ‐invariant. As an application our results are employed to investigate the slice genus of knots. We illustrate examples with arbitrary slice genus for which our lower bound is optimal but all previously known bounds vanish. 57N13, 57N35, 57R95, 57M25

Generalized Seifert surfaces and signatures of colored links

In this paper, we use 'generalized Seifert surfaces' to extend the Levine-Tristram signature to colored links in S 3 . This yields an integral valued function on the μ-dimensional torus, where μ is the number of colors of the link. The case μ = 1 corresponds to the Levine-Tristram signature. We show that many remarkable properties of the latter invariant extend to this μ-variable generalization: it vanishes for achiral colored links, it is 'piecewise continuous', and the places of the jumps are determined by the Alexander invariants of the colored link. Using a 4-dimensional interpretation and the Atiyah-Singer G-signature theorem, we also prove that this signature is invariant by colored concordance, and that it provides a lower bound for the 'slice genus' of the colored link.

Four-dimensional Manifolds

Four-dimensional Manifolds

Ozsváth-Szabó invariants and tight contact three-manifolds, II

Let S 3 (K) be the oriented 3{manifold obtained by rational r{surgery on a knot K S 3 . Using the contact Ozsv ath{Szab o invariants we prove, for a class of knots K containing all the algebraic knots, that S 3 (K) carries positive, tight contact structures for every r 6 g s ( K ) 1, where gs(K) is the slice genus of K . This implies, in particular, that the Brieskorn spheres (2; 3; 4) and (2; 3; 3) carry tight, positive contact structures. As an application of our main result we show that for each m2N there exists a Seifert bered rational homology 3{sphere Mm carrying at least m pairwise non{isomorphic tight, nonllable contact structures.

Ozsváth–Szábo invariants and tight contact three-manifolds I

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 of K. This implies, in particular, that the Brieskorn spheres -Sigma(2,3,4) and -Sigma(2,3,3) carry tight, positive contact structures. As an application of our main result we show that for each m in N there exists a Seifert fibered rational homology 3-sphere M_m carrying at least m pairwise non-isomorphic tight, nonfillable contact structures.

Knot Floer homology and the four-ball genus

We use the knot filtration on the Heegaard Floer complex to define an integer invariant tau(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, tau gives sharp bounds on the four-ball genera of torus knots. As another illustration, we calculate the invariant for several ten-crossing knots.

On the slice genus of links

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 an annulus in the 4-ball.

Homology cobordism and classical knot invariants

In this paper, we define and investigate \( \mathbb{Z}_2 \)-homology cobordism invariants of \( \mathbb{Z}_2 \)-homology 3-spheres which turn out to be related to classical invariants of knots. As an application, we show that many lens spaces have infinite order in the \( \mathbb{Z}_2 \)-homology cobordism group and we prove a lower bound for the slice genus of a knot on which integral surgery yields a given \( \mathbb{Z}_2 \)-homology sphere. We also give some new examples of 3-manifolds which cannot be obtained by integral surgery on a knot.

Quasipositive pretzels

A necessary and sufficient condition for an oriented pretzel surface to be quasipositive yields an estimate for the slice genus of the boundary of an arbitrary oriented pretzel surface.