Narrow Results

In the late 80s, V. Arnold and V. Vassiliev initiated the topological study of the space of real univariate polynomials of a given degree $d$ and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider the spaces ${{𝒫}}_d^{c\mathrm{\Theta }}$ of real monic univariate polynomials of degree $d$ whose real divisors avoid sequences of root multiplicities, taken from a given poset $\mathrm{\Theta }$ of compositions which is closed under certain natural combinatorial operations. In this paper, we concentrate on the fundamental group of ${{𝒫}}_d^{c\mathrm{\Theta }}$ and of some related topological spaces. We find explicit presentations for the groups ${\pi }_1({{𝒫}}_d^{c\mathrm{\Theta }})$ in terms of generators and relations and show that in a number of cases they are free with rank bounded from above by a quadratic function in $d$. We also show that ${\pi }_1({{𝒫}}_d^{c\mathrm{\Theta }})$ stabilizes for $d$ large. The mechanism that generates ${\pi }_1({{𝒫}}_d^{c\mathrm{\Theta }})$ has similarities with the presentation of the braid group as the fundamental group of the space of complex monic degree $d$ polynomials with no multiple roots and with the presentation of the fundamental group of certain ordered configuration spaces over the reals which appear in the work of Khovanov. We further show that the groups ${\pi }_1({{𝒫}}_d^{c\mathrm{\Theta }})$ admit an interpretation as special bordisms of immersions of one-manifolds into the cylinder $ℝ\times S^1$, whose images avoid the tangency patterns from $\mathrm{\Theta }$ with respect to the generators of the cylinder.

We introduce a new relation for high dimensional non-spherical knots, which is motivated by the codimension two surgery theory: a knot is a pull back of another knot if the former is obtained as the inverse image of the latter by a certain degree one map between the ambient spheres. We show that this relation defines a partial order for (2n-1)-dimensional simple fibered knots for n≥3. We also give some related results concerning cobordisms and isotopies of knots together with several important explicit examples.

We reconsider the link homology theory defined by Knovanov in [9] and generalized by Mackaay and Vaz in [15]. With some slight modifications, we describe the theory as a map from the planar algebra of tangles to a planar algebra of complexes of "cobordisms with seams" (actually, a "canopolis"), making it local in the sense of Bar-Natan's local theory of [2].

We show that this "seamed cobordism canopolis" decategorifies to give precisely what you had both hope for and expect: Kuperberg's spider defined in [14]. We conjecture an answer to an even more interesting question about the decategorification of the Karoubi envelope of our cobordism theory.

Finally, we describe how the theory is actually completely computable, and give a detailed calculation of the homology of the (2,n) torus knots.

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.

The purpose of this note is to show that classical cobordism arguments, which go back to the pioneering works of Mandelbaum and Moishezon, provide quick and unified proofs of any knot surgered compact simply-connected 4-manifold $X_K$ becoming diffeomorphic to $X$ after a single stabilization by connected summing with $S^2\times S^2$ or $ℂ\mathbb{P}{}^2\#\overline{ℂ\mathbb{P}}{}^2$, and almost complete decomposability of $X_K$ for many almost completely decomposable $X$, such as the elliptic surfaces.

Our goal is to systematically compute the $ℂP^2$-genus of as many prime knots up to 8-crossings as possible. We obtain upper bounds on the $ℂP^2$-genus via coherent band surgery. We obtain lower bounds by obstructing homological degrees of potential slice discs. The obstructions are pulled from a variety of sources in low-dimensional topology and adapted to $ℂP^2$. There are 27 prime knots and distinct mirrors up to 7-crossings. We now know the $ℂP^2$-genus of all of these knots. There are 64 prime knots and distinct mirrors up to 8-crossings. We now know the $ℂP^2$-genus of all but 6 of these knots, where the $ℂP^2$-genus was not determined explicitly, it was narrowed down to 2 possibilities. As a consequence of this work, we show an infinite family of knots such that the $ℂP^2$-genus of each knot differs from that of its mirror.

In this paper, we study the concordance and cobordism of decompositions associated with defining sequences and we relate them to some invariants of toroidal decompositions and to the cobordism of homology manifolds. These decompositions are often wild Cantor sets and they arise as nested intersections of knotted solid tori. We show that there are at least uncountably many concordance classes of such decompositions in the 3-sphere.

This paper is an introduction to combinatorial knot theory via state summation models for the Jones polynomial and its generalizations. It is also a story about the developments that ensued in relation to the discovery of the Jones polynomial and a remembrance of Vaughan Jones and his mathematics.

The group BSL(k) of boundary cobordism classes of boundary k-string links is defined. An epimorphism from BSL(k) to a group of cobordism classes of matrices is defined. An action of a certain group of pure braids on BSL(k) provides all possible splittings for a given boundary k-link. A necessary and sufficient condition is given for two elements of BSL(k) to have the same closure as an F(k)-link (i.e., a boundary k-link with one of its splittings), up to F(k)-cobordism.

For a map between smooth manifolds, the space of the connected components of its fibers is called the Stein factorization. In our previous paper, we showed that for generic smooth maps, the Stein factorizations are triangulable. As an application, we show that every connected component of a regular fiber is null-cobordant if the top dimensional homology of the Stein factorization vanishes.