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Register a new userTriple linking numbers and surface systems
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We give a refined value group for the collection of triple linking numbers of links in the 3-sphere. Given two links with the same pairwise linking numbers we show that they have the same refined triple linking number collection if and only if the links admit homeomorphic surface systems. Moreover these two conditions hold if and only if the link exteriors are bordant over $B mathbb{Z}^n$, and if and only if the third lower central series quotients $pi/pi_3$ of the link groups are isomorphic preserving meridians and longitudes. We also show that these conditions imply that the link groups have isomorphic fourth lower central series quotients $pi/pi_4$, preserving meridians.
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We establish some new relationships between Milnor invariants and Heegaard Floer homology. This includes a formula for the Milnor triple linking number from the link Floer complex, detection results for the Whitehead link and Borromean rings, and a structural property of the $d$-invariants of surgeries on certain algebraically split links.
In the 1950s Milnor defined a family of higher order invariants generalizing the linking number. Even the first of these new invariants, the triple linking number, has received and fruitful study since its inception. In the case that $L$ has vanishing pairwise linking numbers, this triple linking number gives an integer valued invariant. When the linking numbers fail to vanish, this invariant is only well-defined modulo their greatest common divisor. In recent work Davis-Nagel-Orson-Powell produce a single invariant called the total triple linking number refining the triple linking number and taking values in an abelian group called the total Milnor quotient. They present examples for which this quotient is nontrivial even though none of the individual triple linking numbers are defined. As a consequence, the total triple linking number carries more information than do the classical triple linking numbers. The goal of the present paper is to compute this group and show that when $L$ is a link of at least six components it is non-trivial. Thus, this total triple linking number carries information for every $(nge 6)$-component link, even though the classical triple linking numbers often carry no information.
In this paper we study Turan and Ramsey numbers in linear triple systems, defined as $3$-uniform hypergraphs in which any two triples intersect in at most one vertex. A famous result of Ruzsa and Szemeredi is that for any fixed $c>0$ and large enough $n$ the following Turan-type theorem holds. If a linear triple system on $n$ vertices has at least $cn^2$ edges then it contains a {em triangle}: three pairwise intersecting triples without a common vertex. In this paper we extend this result from triangles to other triple systems, called {em $s$-configurations}. The main tool is a generalization of the induced matching lemma from $aba$-patterns to more general ones. We slightly generalize $s$-configurations to {em extended $s$-configurations}. For these we cannot prove the corresponding Turan-type theorem, but we prove that they have the weaker, Ramsey property: they can be found in any $t$-coloring of the blocks of any sufficiently large Steiner triple system. Using this, we show that all unavoidable configurations with at most 5 blocks, except possibly the ones containing the sail $C_{15}$ (configuration with blocks 123, 345, 561 and 147), are $t$-Ramsey for any $tgeq 1$. The most interesting one among them is the {em wicket}, $D_4$, formed by three rows and two columns of a $3times 3$ point matrix. In fact, the wicket is $1$-Ramsey in a very strong sense: all Steiner triple systems except the Fano plane must contain a wicket.
Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary diagrams. The existence of n-crossing diagrams for every n>1 allows the definition of the n-crossing number. We prove that for any nontrivial, nonsplit link, other than the Hopf link, its triple-crossing number is strictly greater than its quintuple-crossing number.
Given a null-homologous knot $K$ in a rational homology 3-sphere $M$, and the standard infinite cyclic covering $tilde{X}$ of $(M,K)$, we define an invariant of triples of curves in $tilde{X}$, by means of equivariant triple intersections of surfaces. We prove that this invariant provides a map $phi$ on $Al^{otimes 3}$, where $Al$ is the Alexander module of $(M,K)$, and that the isomorphism class of $phi$ is an invariant of the pair $(M,K)$. For a fixed Blanchfield module $(Al,bl)$, we consider pairs $(M,K)$ whose Blanchfield modules are isomorphic to $(Al,bl)$, equipped with a marking, {em i.e.} a fixed isomorphism from $(Al,bl)$ to the Blanchfield module of $(M,K)$. In this setting, we compute the variation of $phi$ under null borromean surgeries, and we describe the set of all maps $phi$. Finally, we prove that the map $phi$ is a finite type invariant of degree 1 of marked pairs $(M,K)$ with respect to null Lagrangian-preserving surgeries, and we determine the space of all degree 1 invariants of marked pairs $(M,K)$ with rational values.
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