Grid Homology for Knots and Links

Ozsváth, Peter; Stipsicz, András I.; Szabó, Zoltán

doi:10.1090/surv/208

Cited by 58 publications

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“…However, in polygonal chains formed by essentially 2D grid diagrams, knotting is stimulated as compared to 3D situation [26]. In our case, the population of non trivial knots starts to exceed 50% at GN 17, and every knot with minimal crossing number ≤ 13 can be represented by grids of size ≤ 12 [20]. This highlights a further advantage of our framework.…”

Section: Resultsmentioning

confidence: 75%

“…Grid diagrams. Grid diagrams are a special kind of knot diagrams, first introduced in [19] and widely used in knot theory (see e.g [20]). Grid diagrams consist of a square, planar, n × n grid (n is a natural number greater than 2), in which 2n markings are placed, corresponding to vertices of a piecewise linear curve representing a given knot.…”

Section: Resultsmentioning

confidence: 99%

“…Although DNA forms a double helix, for such topological aspects as DNA knots, DNA helicity can be frequently neglected and the formed knot can be recognised from a projection of the axial path of analysed DNA molecules [1]. Such projections can be conveniently represented as grid diagrams [20] (see Figures 1 and 2). Grid diagrams encode the information of which arc is over-passing and which is under-passing at each crossing (see Figure 1, (A)).…”

Section: Methodsmentioning

confidence: 99%

“…Theorem. [20,Ch.3] Any link L can be represented by a grid diagram. In this setting, strand passages ( i.e.…”

Section: Definitionmentioning

confidence: 99%

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Grid diagrams as tools to investigate knot spaces and topoisomerase-mediated simplification of DNA topology

et al. 2020

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Grid diagrams with their relatively simple mathematical formalism provide a convenient way to generate and model projections of various knots. It has been an open question whether these 2D diagrams can be used to model a complex 3D process such as the topoisomerasemediated preferential unknotting of DNA molecules. We model here topoisomerase-mediated passages of double-stranded DNA segments through each other using the formalism of grid diagrams. We show that this grid diagram-based modelling approach captures the essence of the preferential unknotting mechanism, based on topoisomerase selectivity of hooked DNA juxtapositions as the sites of intersegmental passages. We show that grid diagram-based approach provide an important, new and computationally convenient framework for investigating entanglement in biopolymers. arXiv:1909.05937v1 [q-bio.BM]

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Section: Resultsmentioning

confidence: 75%

Section: Resultsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

“…Theorem. [20,Ch.3] Any link L can be represented by a grid diagram. In this setting, strand passages ( i.e.…”

Section: Definitionmentioning

confidence: 99%

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Grid diagrams as tools to investigate knot spaces and topoisomerase-mediated simplification of DNA topology

et al. 2020

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“…Based on the Gluing Theorem one can then use the maps coming from these elementary computations to express the differential of knot Floer homology. This leads to a combinatorial formulation of knot Floer homology (see Section section 7.5 below) alternative to grid homology [5]. 7.1.…”

Section: The Bimodules Of Some Elementary Configurationsmentioning

confidence: 99%

An introduction to knot Floer homology and curved bordered algebras

Alfieri¹,

Dyke

2020

Period Math Hung

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We survey Ozsváth-Szabó's bordered approach to knot Floer homology. After a quick introduction to knot Floer homology, we introduce the relevant algebraic concepts (A ∞ -modules, type D-structures, box tensor product, etc.), we discuss partial Kauffman states, the construction of the boundary algebra, and sketch Ozsváth and Szabó's analytic construction of the type D-structure associated to an upper diagram. Finally we give an explicit description of the structure maps of the DA-bimodules of some elementary partial diagrams. These can be used to perform explicit computations of the knot Floer differential of any knot in S 3 . The boundary DGAs B(n, k) and A(n, k) of [6] are replaced here by an associative algebra C(n). These are the notes of two lecture series delivered by Peter Ozsváth and Zoltán Szabó at Princeton University during the summer of 2018.A powerful paradigm in computer science is the so called Divide et Impera. It consists of dividing a complicated problem into smaller subproblems in order to solve them individually. In topology this translates to the cut and paste philosophy. More specifically: suppose that one wants to compute an invariant D of some space X, then one can divide X into pieces P 1 , . . . , P N and can compute some sort of related invariant I for the pieces, and develop some gluing formula expressing D(X) as a function of I(P 1 ), . . . , I(P N ). A typical example is when D(X) is the fundamental group: in this case the invariants I(P i ) are some inclusion maps, and the gluing principle is the Seifert-van Kampen Theorem.In four-manifold topology, cut and paste techniques to compute Seiberg-Witten invariants were pioneered in [2] and eventually led to the definition of the Monopole Floer homology groups by Kronheimer and Mrowka [3]. In the case of the Heegaard Floer three-manifold groups an implementation of the cut and paste approach first appeared in [4]. In this work Lipshitz, Ozsváth and Thurston associate to three manifolds with boundary Y 1 and Y 2 a type D-structure CF D(Y 1 ) and awhere, whatever these algebraic structures are, is some sort of tensor product operation that takes as input a type D-structure and an A ∞ -module, and gives back a chain complex.Ozsváth and Szabó developed a similar construction in the setting of knot Floer homology [6]. Given a knot K ⊂ R 3 one can split it into two parts by means of a two-plane z = t. Denote by K [t,+∞) = K ∩ {z ≥ t} the portion of K lying above the plane z = t. Similarly denote by K (−∞,t] = K ∩ {z ≤ t} the part lying below it. In [6] Ozsváth and Szabó associate to K [t,+∞) and K (−∞,t] a type D-structure DF K(K [t,+∞) ) and an A ∞ -module AF K(K (−∞,t]) ) so that CF K(K) = AF K(K (−∞,t]) ) DF K(K [t,+∞) ) .More generally, they develop some sort of Morse theoretic approach one can apply to compute knot Floer homology: given t 1 < t 2 they associate toThe scope of these lecture notes is to survey the algebraic language of A ∞ -modules, type D-and DA-structures, and sketch Ozsváth and Szabó's recent construction. We will only a...

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Knot Floer homology and relative adjunction inequalities

Hedden

Raoux

2022

Sel. Math. New Ser.

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Grid Homology for Knots and Links

Cited by 58 publications

References 0 publications

Grid diagrams as tools to investigate knot spaces and topoisomerase-mediated simplification of DNA topology

Grid diagrams as tools to investigate knot spaces and topoisomerase-mediated simplification of DNA topology

An introduction to knot Floer homology and curved bordered algebras

Knot Floer homology and relative adjunction inequalities

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