Donaldson-Thomas invariants, torus knots, and lattice paths

In this paper we find and explore the correspondence between quivers, torus knots, and combinatorics of counting paths. Our first result pertains to quiver representation theory -- we find explicit formulae for classical generating functions and Donaldson-Thomas invariants of an arbitrary symmetric quiver. We then focus on quivers corresponding to $(r,s)$ torus knots and show that their classical generating functions, in the extremal limit and framing $rs$, are generating functions of lattice paths under the line of the slope $r/s$. Generating functions of such paths satisfy extremal A-polynomial equations, which immediately follows after representing them in terms of the Duchon grammar. Moreover, these extremal A-polynomial equations encode Donaldson-Thomas invariants, which provides an interesting example of algebraicity of generating functions of these invariants. We also find a quantum generalization of these statements, i.e. a relation between motivic quiver generating functions, quantum extremal knot invariants, and $q$-weighted path counting. Finally, in the case of the unknot, we generalize this correspondence to the full HOMFLY-PT invariants and counting of Schr\"oder paths.Comment: 45 pages. Corrected typos in new versio

The change of feedback invariants under one row perturbation

AbstractIn this paper we completely characterize possible feedback invariants of a rectangular matrix under small additive perturbations on one of its rows

Parasupersymmetric Quantum Mechanics of Order 3 and a Generalized Witten Index

In this paper we generalize the Rubakov-Spiridonov parasupersymmetry algebra to the order 3 case. We also generalize the notion of the Witten index, and we provide a class of models satisfying our parasupersymmetry algebra. Finally, we show that there is a correspondence between the Hamiltonian and the index in our class of models

Homological thickness and stability of torus knots

In this paper we show that the non-alternating torus knots are homologically thick, i.e. that their Khovanov homology occupies at least three diagonals. Furthermore, we show that we can reduce the number of full twists of the torus knot without changing certain part of its homology, and consequently, we show that there exists stable homology of torus knots conjectured by Dunfield, Gukov and Rasmussen in \cite{dgr}. Since our main tool is the long exact sequence in homology, we have applied our approach in the case of the Khovanov-Rozansky ($sl(n)$) homology, and thus obtained analogous stability properties of $sl(n)$ homology of torus knots, also conjectured in \cite{dgr}.Comment: 24 pages, expanded Section

Quadruply-graded colored homology of knots

We conjecture the existence of four independent gradings in colored HOMFLYPT homology, and make qualitative predictions of various interesting structures and symmetries in the colored homology of arbitrary knots. We propose an explicit conjectural description for the rectangular colored homology of torus knots, and identify the new gradings in this context. While some of these structures have a natural interpretation in the physical realization of knot homologies based on counting supersymmetric configurations (BPS states, instantons, and vortices), others are completely new. They suggest new geometric and physical realizations of colored HOMFLYPT homology as the Hochschild homology of the category of branes in a Landau–Ginzburg B-model or, equivalently, in the mirror A-model. Supergroups and supermanifolds are surprisingly ubiquitous in all aspects of this work

Postkrizni pravci kretanja zaposlenosti u Srbiji

Svetska ekonomska kriza iz 2008. godine ima negativne posledice po privredna kretanja u Srbiji. Zbog smanjene tražnje, kako na domaćim, tako i na stranim tržištima, dolazi isprva do drastičnog pada potrošnje i trgovine sa inostranstvom. Usporena privredna aktivnost, smanjena tražnja, pesimistička očekivanja, inflatorni pritisak, nestabilna i nagla deprecijacija domaće valute, smanjen nivo priliva stranih direktnih investicija, kao i povećanje javnog duga, obeležili su period između 2008. i 2010. godine. Kroz mehanizam smanjenih investicija, umanjenu inostranu tražnju za domaćim proizvodima, kao i kroz složeniji pristup finansijskim sredstvima, svetska ekonomska kriza se preliva u Srbiju. Domaće posledice: manji budžetski prihodi, povećani socijalni izdaci, snažna deprecijacija u odnosu na evro, nejedinstvena politika fiskalnih i monetarnih vlasti, što је sve imalo uticaja na veoma strmi pad zaposlenosti i standarda. U ovom radu se osvrćemo na uzroke krize u Srbiji, koja nije u potpunosti uvezena iz inostranstva, već poseduje i odlike uzrokovane unutar srpske privede. Potom, daćemo sektorski prikaz dinamike zaposlenosti u periodu posle 2000. godine, sa analizom stanja u onim sektorima koji su ključni za zaposlenost

BPS states, knots and quivers

We argue how to identify the supersymmetric quiver quantum mechanics description of BPS states, which arise in string theory in brane systems representing knots. This leads to a surprising relation between knots and quivers: to a given knot, we associate a quiver, so that various types of knot invariants are expressed in terms of characteristics of a moduli space of representations of the corresponding quiver. This statement can be regarded as a novel type of categorification of knot invariants, and among its various consequences we find that Labastida-Mariño-Ooguri-Vafa (LMOV) invariants of a knot can be expressed in terms of motivic Donaldson-Thomas invariants of the corresponding quiver; this proves integrality of LMOV invariants (once the corresponding quiver is identified), conjectured originally based on string theory and M-theory arguments

Knots-quivers correspondence

We introduce and explore the relation between knot invariants and quiver representation theory, which follows from the identification of quiver quantum mechanics in D‑brane systems representing knots. We identify various structural properties of quivers associated to knots, and identify such quivers explicitly in many examples, including some infinite families of knots, all knots up to 6 crossings, and some knots with thick homology. Moreover, based on these properties, we derive previously unknown expressions for colored HOMFLY‑PT polynomials and superpolynomials for various knots. For all knots, for which we identify the corresponding quivers, the LMOV conjecture for all symmetric representations (i.e. integrality of relevant BPS numbers) is automatically proved