我们分析了具有少量负交叉的图的链接的属性。我们表明，如果一个非平凡的链接有一个图，所有的交叉点，除了可能是一个积极的，那么签名的链接是负的。如果一个链接图有两个负的交叉点，我们表明，该链接的签名是非正的左手Hopf链接（可能平凡的组件）除外。我们还刻画了具有签名零的链路和具有两个负交叉的图。特别是，我们表明，如果一个非平凡的结有一个图有两个负交叉，然后签名的结是负的，除非结是一个扭曲结负扣。我们完全确定所有平凡的链接图有两个或更少的负交叉。对于一个有三个负交叉的纽结图，除了左手三叶结之外，纽结的签名都是非正的。这些结果推广了L. Rudolph，T. Cochran，E. Gompf，P. Traczyk，and J. H. Przytycki，解决了[P-2]的猜想5，并部分回答了[Co-G]的问题2.8关于纽结支配三叶结或平凡纽结的问题。我们还描述了所有解开的一号正结。
英文摘要：
We analyze properties of links which have diagrams with a small number of negative crossings. We show that if a nontrivial link has a diagram with all crossings positive except possibly one, then the signature of the link is negative. If a link diagram has two negative crossings, we show that the signature of the link is nonpositive with the exception of the left-handed Hopf link (with possible trivial components). We also characterize those links which have signature zero and diagrams with two negative crossings. In particular, we show that if a nontrivial knot has a diagram with two negative crossings then the signature of the knot is negative, unless the knot is a twist knot with negative clasp. We completely determine all trivial link diagrams with two or fewer negative crossings. For a knot diagram with three negative crossings, the signature of the knot is nonpositive except the left-handed trefoil knot. These results generalize those of L. Rudolph, T. Cochran, E. Gompf, P. Traczyk, and J. H. Przytycki, solve Conjecture 5 of [P-2], and give a partial answer to Problem 2.8 of [Co-G] about knots dominating the trefoil knot or the trivial knot. We also describe all unknotting number one positive knots.
As an extension of positive or almost positive diagrams and links, we introduce a notion of successively almost positive diagrams and links, and good successively almost positive diagrams and links. We review various properties of positive links or almost positive links, and explain how they can be extended to (good) successively almost positive links. Our investigation also leads to an improvement of known results of positive or almost positive links.
As an extension of positive and almost positive diagrams and links, we study two classes of links we call successively almost positive and weakly successively almost positive links. We prove various properties of polynomial invariants and signatures of such links, extending previous results or answering open questions about positive or almost positive links. We discuss their minimal genus and fibering property and for the latter prove a fibering extension of Scharlemann-Thompson's theorem (valid for general links).
Relations between the Atiyah-Patodi-Singer rho invariant and signatures of links have been known for a long time, but they were only partially investigated. In order to explore them further, we develop a versatile cut-and-paste formula for the rho invariant, which allows us to manipulate manifolds in a convenient way. With the help of this tool, we give a description of the multivariable signature of a link $L$ as the rho invariant of some closed $3$-manifold $Y_L$ intrinsically associated to $L$. We study then the rho invariant of the manifolds obtained by Dehn surgery on a $L$ along integer …
We construct an infinite family of links which are both almost alternating and quasi-alternating from a given either almost alternating diagram representing a quasi-alternating link, or connected and reduced alternating tangle diagram. To do that we use what we call a dealternator extension which consists in replacing the dealternator by a rational tangle extending it. We note that all not alternating and quasi-alternating Montesinos links can be obtained in that way. We check that all the obtained quasi-alternating links satisfy Conjecture 3.1 of Qazaqzeh et al. (JKTR 22 (06), 2013), that is …
This paper studies existence of $n=4k (k>1)$ dimensional simply-connected closed almost complex manifold with Betti number $ b_i=0$ except $i=0, n/2, n$. We characterize all the rational cohomology rings of such manifolds and show they must have even Euler characteristic and even signature, which is to say the middle Betti number $b_{n/2}$ must be even. Parallel to the author's earlier work on realizing rational cohomology ring by smooth closed manifolds, we state and prove Sullivan's rational surgery realization theorem for almost complex manifold and demonstrate its application in our contex…
We prove that any link admitting a diagram with a single negative crossing is strongly quasipositive. This answers a question of Stoimenow's in the (strong) positive. As a second main result, we give simple and complete characterizations of link diagrams with quasipositive canonical surface (the surface produced by Seifert's algorithm). As applications, we determine which prime knots up to 13 crossings are strongly quasipositive, and we confirm the following conjecture for knots that have a canonical surface realizing their genus: a knot is strongly quasipositive if and only if the Bennequin i…
We derive a lower bound for all Levine-Tristram signatures of positive braid links, linear in terms of the first Betti number. As a consequence, we obtain upper and lower bounds on the ratio of fixed pairs of Levine-Tristram signature invariants, valid uniformly on all positive braid monoids.
Using the Gordon-Litherland pairing, one can define invariants (signature, nullity, determinant) for ${\mathbb Z}/2$ null-homologous links in thickened surfaces. In this paper, we study the concordance properties of these invariants. For example, if $K \subset Σ\times I$ is ${\mathbb Z}/2$ null-homologous and slice, we show that its signatures vanish and its determinants are perfect squares. These statements are derived from a cobordism result for closed unoriented surfaces in certain 4-manifolds. The Brown invariants are defined for ${\mathbb Z}/2$ null-homologous links in thickened surfaces.…
