Electrostatic description of 3d $$ \mathcal{N} $$ = 4 linear quivers

We present the holographic dual for the strongly coupled, low energy dynamics of balanced$$ \mathcal{N} $$ N = 4 field theories in (2 + 1) dimensions. The infinite family of Type IIB backgrounds with AdS4× S2× S2 factors is described in terms of a Laplace problem with suitable boundary conditions. The system describes an array of D3, NS5 and D5 branes. We study various aspects of these Hanany-Witten set-ups (number of branes, linking numbers, dimension of the Higgs and Coulomb branches) and encode them in holographic calculations. A generic expression for the Free Energy/Holographic Central Charge is derived. These quantities are then calculated explicitly in various general examples. We also discuss how Mirror Symmetry is encoded in our Type IIB backgrounds. The connection with previous results in the bibliography is made.

Linking numbers, quandles and groups

We introduce a quandle invariant of classical and virtual links, denoted by [Formula: see text]. This quandle has the property that [Formula: see text] if and only if the components of [Formula: see text] and [Formula: see text] can be indexed in such a way that [Formula: see text], [Formula: see text] and for each index [Formula: see text], there is a multiplier [Formula: see text] that connects virtual linking numbers over [Formula: see text] in [Formula: see text] to virtual linking numbers over [Formula: see text] in [Formula: see text]: [Formula: see text] for all [Formula: see text]. We also extend to virtual links a classical theorem of Chen, which relates linking numbers to the nilpotent quotient [Formula: see text].

Linking Numbers in Three-Manifolds

Let M be a connected, closed, oriented three-manifold and K, L two rationally null-homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number between K and L in terms of a presentation of M as an irregular dihedral three-fold cover of $$S^3$$ S 3 branched along a knot $$\alpha \subset S^3$$ α ⊂ S 3 . Since every closed, oriented three-manifold admits such a presentation, our results apply to all (well-defined) linking numbers in all three-manifolds. Furthermore, ribbon obstructions for a knot $$\alpha $$ α can be derived from dihedral covers of $$\alpha $$ α . The linking numbers we compute are necessary for evaluating one such obstruction. This work is a step toward testing potential counter-examples to the Slice-Ribbon Conjecture, among other applications.

Trail Making Test Performance Using a Touch-Sensitive Tablet: Behavioral Kinematics and Electroencephalography

The Trail Making Test (TMT) is widely used to probe brain function and is performed with pen and paper, involving Parts A (linking numbers) and B (alternating between linking numbers and letters). The relationship between TMT performance and the underlying brain activity remains to be characterized in detail. Accordingly, sixteen healthy young adults performed the TMT using a touch-sensitive tablet to capture enhanced performance metrics, such as the speed of linking movements, during simultaneous electroencephalography (EEG). Linking and non-linking periods were derived as estimates of the time spent executing and preparing movements, respectively. The seconds per link (SPL) was also used to quantify TMT performance. A strong effect of TMT Part A and B was observed on the SPL value as expected (Part B showing increased SPL value); whereas the EEG results indicated robust effects of linking and non-linking periods in multiple frequency bands, and effects consistent with the underlying cognitive demands of the test.

Exchangeability and Non-Conjugacy of Braid Representatives

We obtain some fairly general conditions on the linking numbers and geometric properties of a link, under which it has infinitely many conjugacy classes of [Formula: see text]-braid representatives if and only if it has one admitting an exchange move. We investigate a symmetry pattern of indices of conjugate iterated exchanged braids. We then develop a test based on the Burau matrix showing examples of knots admitting no minimal exchangeable braids, admitting non-minimal non-exchangeable braids, and admitting both minimal exchangeable and minimal non-exchangeable braids. This in particular proves that conjugacy, exchange moves and destabilization do not suffice to simplify braid representatives of a general link.

Topological interpretation of color exchange invariants: hexagonal lattice on a torus

We explain a correspondence between some invariants in the dynamics of color exchange in the coloring problem of a 2d regular hexagonal lattice, which are polynomials of winding numbers, and linking numbers in 3d. One invariant is visualized as linking of lines on a special surface with Arf-Kervaire invariant one, and is interpreted as resulting from an obstruction to transform the surface into its chiral image with special continuous deformations. We also consider additional constraints on the dynamics and see how the surface is modified.