Narrow Results

A spatial refinement of Bar-Natan homology is given, that is, for any link diagram $D$ we construct a CW-spectrum ${{𝒳}}_{BN}(D)$ whose reduced cellular cochain complex gives the Bar-Natan complex of $D$. The stable homotopy type of ${{𝒳}}_{BN}(D)$ is a link invariant and is described as the wedge sum of the “canonical sphere spectra”. We conjecture that the quantum filtration of Bar-Natan homology also lifts to the spatial level, and that it leads us to a cohomotopical refinement of the $s$-invariant.

We use recently introduced Rasmussen invariant to find knots that are topologically locally-flatly slice but not smoothly slice. We note that this invariant can be used to give a combinatorial proof of the slice-Bennequin inequality. Finally, we compute the Rasmussen invariant for quasipositive knots and show that most of our examples of non-slice knots are not quasipositive and, to the best of our knowledge, were previously unknown.

By using Rasmussen's s-invariant derived from Khovanov homology, we give a combinatorial proof for the existence of countably many Casson handles.

We compute integer valued knot concordance invariants of a family of general pretzel knots if the invariants are equal to the negative values of signatures for alternating knots. Examples of such invariants are Rasmussen s-invariants and twice Ozsváth–Szabó knot Floer homology τ-invariants. We use the crossing change inequalities of Livingston and the fact that pretzel knots are almost alternating. As a consequence, for the family of pretzel knots given in this paper, s-invariants are twice τ-invariants.

We introduce a new class of links for which we give a lower bound for the slice genus g_*, using the generalized Rasmussen invariant. We show that this bound, in some cases, allows one to compute g_* exactly; in particular, we compute g_* for torus links. We also study another link invariant: the strong slice genus . Studying the behavior of a specific type of cobordisms in Lee homology, a lower bound for is also given.

We present a large family of knots for which the Rasmussen $s$-invariants of arbitrary satellites do not detect sliceness. This answers a question of Hedden. The proof hinges on work of Kronheimer–Mrowka and Cochran–Harvey–Horn.

The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov–Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies as well to classical knots and classical knot cobordisms. We give an alternate formulation for the Manturov definition [34] of Khovanov homology [25], [26] for virtual knots and links with arbitrary coefficients. This approach uses cut loci on the knot diagram to induce a conjugation operator in the Frobenius algebra. We use this to show that a large class of virtual knots with unit Jones polynomial is non-classical, proving a conjecture in [20] and [10]. We then discuss the implications of the maps induced in the aforementioned theory to the universal Frobenius algebra [27] for virtual knots. Next we show how one can apply the Karoubi envelope approach of Bar-Natan and Morrison [3] on abstract link diagrams [17] with cross cuts to construct the canonical generators of the Khovanov–Lee homology [30]. Using these canonical generators we derive a generalization of the Rasmussen invariant [39] for virtual knot cobordisms and generalize Rasmussen’s result on the slice genus for positive knots to the case of positive virtual knots. It should also be noted that this generalization of the Rasmussen invariant provides an easy to compute obstruction to knot cobordisms in $S_g\times I\times I$ in the sense of Turaev [42].

These five lectures — written for a summer school in 2006 — provide an introduction to Khovanov homology covering the basic definitions, important properties, a number of variants and some applications. At the end of each lecture the reader is referred to the relevant literature for further reading.

This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant under certain forms of labeled cobordism and study a number of examples in relation to these phenomena. Information on determinations of the four-ball genus of some virtual knots is obtained by via the affine index polynomial in conjunction with results on the genus of positive virtual knots using joint work with Dye and Kaestner.

We give a description of Rasmussen’s $s$-invariant from the divisibility of Lee’s canonical class. More precisely, given any link diagram $D$, for any choice of an integral domain $R$ and a non-zero, non-invertible element $c\in R$, we define the $c$-divisibility $k_c(D)$ of Lee’s canonical class of $D$, and prove that a combination of $k_c(D)$ and some elementary properties of $D$ yields a link invariant ${\bar{s}}_c$. Each ${\bar{s}}_c$ possesses properties similar to $s$, which in particular reproves the Milnor conjecture. If we restrict to knots and take $(R,c)=(\mathbb{Q}[h],h)$, then our invariant coincides with $s$.