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Register a new userOn mod 3 triple Milnor invariants and triple cubic residue symbols in the Eisenstein number field
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We introduce mod 3 triple Milnor invariants and triple cubic residue symbols for certain primes of the Eisenstein number field $mathbb{Q}(sqrt{-3})$, following the analogies between knots and primes. Our triple symbol generalizes both the cubic residue symbol and R{e}deis triple symbol, and describes the decomposition law of a prime in a mod 3 Heisenberg extension of degree 27 over $mathbb{Q}(sqrt{-3})$ with restricted ramification, which we construct concretely in the form similar to R{e}deis dihedral extension over $mathbb{Q}$. We also give a cohomological interpretation of our symbols by triple Massey products in Galois cohomology.
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The use of quadratic residues to construct matrices with specific determinant values is a familiar problem with connections to many areas of mathematics and statistics. Our research has focused on using cubic residues to construct matrices with interesting and predictable determinants.
The subject of this work is a three-dimensional topological field theory with a non-semisimple group of gauge symmetry with observables consisting in the holonomies of connections around three closed loops. The connections are a linear combination of gauge potentials with coefficients containing a set of one-dimensional scalar fields. It is checked that these observables are both metric independent and gauge invariant. The gauge invariance is achieved by requiring non-trivial gauge transformations in the scalar field sector. This topological field theory is solvable and has only a relevant amplitude which has been computed exactly. From this amplitude it is possible to isolate a topological invariant which is Milnors triple linking invariant. The topological invariant obtained in this way is in the form of a sum of multiple contour integrals. The contours coincide with the trajectories of the three loops mentioned before. The introduction of the one-dimensional scalar field is necessary in order to reproduce correctly the particular path ordering of the integration over the contours which is present in the triple Milnor linking coefficient. This is the first example of a local topological gauge field theory that is solvable and can be associated to a topological invariant of the complexity of the triple Milnor linking coefficient.
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.
In a recent work, Jungnickel, Magliveras, Tonchev, and Wassermann derived an overexponential lower bound on the number of nonisomorphic resolvable Steiner triple systems (STS) of order $v$, where $v=3^k$, and $3$-rank $v-k$. We develop an approach to generalize this bound and estimate the number of isomorphism classes of STS$(v)$ of rank $v-k-1$ for an arbitrary $v$ of form $3^kT$.
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