This note has an experimental nature and contains no new theorems. We introduce certain moves for classical knot diagrams that for all the very many examples we have tested them on give a monotonic complete simplification. A complete simplification of a knot diagram D is a sequence of moves that transform D into a diag…
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48 results for “diagram simplification”
In the present paper a criteria for a rectangular diagram to admit a simplification is given in terms of Legendrian knots. It is shown that there are two types of simplifications which are mutually independent in a sense. A new proof of the monotonic simplification theorem for the unknot is given. It is shown that a mi…
In a recent work "Arc-presentation of links: Monotonic simplification" Ivan Dynnikov showed that each rectangular diagram of the unknot, composite link, or split link can be monotonically simplified into a trivial, composite, or split diagram, respectively. The following natural question arises: Is it always possible t…
The second author previously discussed how classical complexity separation conjectures, we call them "axioms", have implications in three manifold topology: polynomial length stings of operations which preserve certain Jones polynomial evaluations cannot produce exponential simplifications of link diagrams. In this pap…
We describe the Polyak-Viro arrow diagram formulas for the coefficients of the Conway polynomial. As a consequence, we obtain the Conway polynomial as a state sum over some subsets of the crossings of the knot diagram. It turns out to be a simplification of a special case of Jaeger's state model for the HOMFLY polynomi…
RL pipeline simplifies knot diagrams, including very hard unknots.
problem Simplifying complex knot diagrams, especially very hard unknots.
method Reinforcement learning for move proposals and heuristic navigation of Reidemeister moves.
result Trained agent simplifies diagrams, including a link to a three-step unknotting process.
Simplified plat diagrams for unlink without stabilization.
problem Equivalence of plats without stabilization for unlink.
method Introducing pocket and flip moves to simplify plats.
result Simplified plat diagrams for unlink using pocket and flip moves.
On Free Knots and Linksmath.GT
Both classical and virtual knots arise as formal Gauss diagrams modulo some abstract moves corresponding to Reidemeister moves. If we forget about both over/under crossings structure and writhe numbers of knots modulo the same Reidemeister moves, we get a dramatic simplification of virtual knots, which kills all classi…
Goussarov, Polyak, and Viro proved that finite type invariants of knots are ``finitely multi-local'', meaning that on a knot diagram, sums of quantities, defined by local information, determine the value of the knot invariant. The result implies the existence of Gauss diagram combinatorial formulas for finite type inva…
Simplified KR polynomial for bipartite links reduces to tensor products of vector spaces.
problem Complexity reduction of Khovanov-Rozansky polynomial for bipartite links.
method Local reduction of matrix factorizations to planar cycles and simplification to vector spaces.
result KR polynomial for bipartite links simplifies to tensor products of vector spaces.
A branched covering surface-knot over an oriented surface-knot is a surface-knot in the form of a branched covering over . A branched covering surface-knot over is presented by a graph called a chart on a surface diagram of . For a branched covering surface-knot, an addition of 1-handles equipped with cha…
The list of knots with up to 10 crossings is commonly referred to as the Rolfsen Table. This paper presents a way to generate the Rolfsen table in a simple, clear, and reproducible manner. The methods we use are similar to those used by J. Hoste, M. Thistlethwaite, and J. Weeks in [1]. The difference between our method…
Software simplifies triangulations of 4-manifolds, revealing exotic structures.
problem Understanding smooth 4-manifolds from discrete and algorithmic perspectives.
method New software tools, including an algorithm for triangulations from Kirby diagrams and a heuristic for simplification.
result Presented new triangulations of exotic pairs, corks, and plugs, including the smallest known K3 surface.
This paper defines a new invariant of virtual knots and links that we call the extended bracket polynomial, and denote by <<K>> for a virtual knot or link K. This invariant is a state summation over bracket states of the oriented diagram for K. Each state is reduced to a virtual 4-regular graph in the plane and the pol…
New method for simplifying complex 4D shapes with boundaries.
problem Complexity reduction in 4D shape analysis.
method Introducing pseudo-trisections and pseudo-bridge trisections.
result Existence and uniqueness of pseudo-trisections established.
A branched covering surface-knot is a surface-knot in the form of a branched covering over an oriented surface-knot , where we include the case when the covering has no branch points. A branched covering surface-knot is presented by a graph called a chart on a surface diagram of . We can simplify a branched cover…
Develops a framework for stuck knots with rigid constraints and invariants.
problem Rigidity constraints on knot diagrams restrict allowable isotopies.
method Formalizes stuck crossings as rigid configurations, introduces unstick move, and constructs invariants.
result Rigidity contributes independent information even when knot type remains fixed.
We establish a full principle (close, relative, parametric) for the simplification of singularities of Lagrangian and Legendrian fronts. More precisely, we prove that if there is no homotopy theoretic obstruction to simplifying the singularities of tangency of a Lagrangian or Legendrian submanifold with respe…
We prove that any arc-presentation of the unknot admits a monotonic simplification by elementary moves; this yields a simple algorithm for recognizing the unknot. We obtain similar results for split links and composite links.
Natural gradient simplification for deep learning networks.
problem Efficiency in training deep Bayesian networks.
method Analysis of two geometries of Fisher information matrix and development of a method to simplify natural gradient for the second geometry.
result A method to simplify natural gradient for deep networks using an auxiliary recognition model.
Simplified Khovanov-Rozansky operators for link invariants.
problem Complexity in Khovanov-Rozansky knot invariants.
method Substituted matrix factorization with simple operators and conjugations.
result Global reductions for simplifying link invariants.
It is generally accepted that all models are wrong -- the difficulty is determining which are useful. Here, a useful model is considered as one that is capable of combining data and expert knowledge, through an inversion or calibration process, to adequately characterize the uncertainty in predictions of interest. This…
New diagonal move simplifies knots and links efficiently.
problem Efficiently unknotting knots and links.
method Introduces diagonal move, proving its effectiveness for classical and welded knots.
result Diagonal move reduces any knot or link to the unknot or unlink with fewer operations.
Recent work on stable minimal hypersurface singularities.
problem Understanding singularities of stable minimal hypersurfaces.
method Simplifications of technical discussion in previous work.
result Simplified approach to analyzing hypersurface singularities.
Unwrapping The Black Box of Deep ReLU Networks: Interpretability, Diagnostics, and Simplificationcs.LG
This paper simplifies deep ReLU networks into local linear models for better interpretability.
problem Limited transparency and interpretability of deep neural networks, especially ReLU networks.
method Local linear representation and equivalent set of local linear models (LLMs).
result Simplified deep ReLU networks for better interpretability and diagnostics.
Simplifies residual flows to make flow-based modeling more practical.
problem Extremely high computational cost of residual flows limits their applicability.
method Introduces Quasi-Autoregressive (QuAR) approach to residual flows.
result Significantly reduces compute time and memory requirements for flow-based modeling.
Algorithm simplifies Khovanov homology computations for 4-strand torus links.
problem Computing Khovanov homology of 4-strand torus links.
method Algorithmic Morse theoretic simplifications postponed until the end.
result Non-trivial Khovanov homology groups in all homological degrees for 4-strand torus links.
Title Redactedcs.CY
New method simplifies data analysis.
problem Complex data analysis challenges.
method Innovative algorithm for data simplification.
result Significant reduction in analysis time.
AST provides a method to validate safe autonomy without unsafe simplifications.
problem Validation of safe autonomy in complex systems.
method Adaptive Stress Testing (AST) approach.
result AST can find failures without unsafe simplifications.
Random knotting in very long off-lattice self-avoiding polygonscond-mat.stat-mech
Study on knotting in very long polymer chains, finding Poisson distribution for prime knot types.
problem Understanding knotting in very long polymer chains.
method Generated and analyzed polygons of size using tree data structure and pivot algorithm. Used new knot diagram simplification and invariant-free classification.
result Number of prime summands of knot type in a random -gon is well described by a Poisson distribution.
Two smooth manifolds M and N are called R-diffeomorphic if their product with the real line are diffeomorphic. We consider the following simplification problem: does R-diffeomorphism imply diffeomorphism or homeomorphism? For compact manifolds, analysis of this problem relies on some of the main achievements of the the…
Simplifies RF predictions by focusing on a subset of nearest neighbors.
problem Improving interpretability and performance of RF-based forecast distributions.
method Sparsifying RF-based forecast distributions by focusing on a small subset of nearest neighbors.
result Simplified RF predictions can be similar to or exceed original ones in forecasting performance.
We define a new combinatorial complex computing the hat version of link Floer homology over Z/2Z, which turns out to be significantly smaller than the Manolescu-Ozsvath-Sarkar one.
Framed flow categories were introduced by Cohen-Jones-Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW-complex with one cell for each object of the category. The idea is that the Floer invariant should take the form of the stable homotopy type of the r…
We propose a simple yet effective technique to simplify the training and the resulting model of neural networks. In back propagation, only a small subset of the full gradient is computed to update the model parameters. The gradient vectors are sparsified in such a way that only the top-k elements (in terms of magnitude…
Simplifying triangulationsmath.GT
We give a new algorithm to simplify a given triangulation with respect to a given curve. The simplification uses flips together with powers of Dehn twists in order to complete in polynomial time in the bit-size of the curve.
Eliashberg simplifies singularities in geometry.
problem Complex singularities in geometric structures.
method Philosophy of the h-principle and simplification techniques.
result Simplified understanding of singularities in geometry.
Search-based methods for hard combinatorial optimization are often guided by heuristics. Tuning heuristics in various conditions and situations is often time-consuming. In this paper, we propose NeuRewriter that learns a policy to pick heuristics and rewrite the local components of the current solution to iteratively i…
We describe a simplification of Donaldson's arguments for the construction of symplectic hypersurfaces or Lefschetz pencils that makes it possible to avoid any reference to Yomdin's work on the complexity of real algebraic sets.
Advertising effects in Sznajd marketing modelcond-mat.stat-mech
The traditional Sznajd model, as well as its Ochrombel simplification for opinion spreading, are applied to marketing with the help of advertising. The larger the lattice is the smaller is the amount of advertising needed to convince the whole market
Simplified Khovanov-Rozansky calculus for bipartite knots.
problem Complexity in calculating superpolynomials for knots.
method Bipartite calculus generalizes Khovanov-Rozansky calculus for a restricted class of knots.
result Simplification of Khovanov-Rozansky polynomials for bipartite knots.
The paper refines transformations of lattice diagrams and introduces dotted diagrams.
problem Investigating transformations and deformations of lattice diagrams and their associated dotted diagrams.
method Introducing dotted diagrams and investigating deformations of these diagrams, relating them to transformations of lattice diagrams.
result Refined results on the relation between deformations of admissible dotted diagrams and transformations of lattice diagrams.
Morphology in unbalanced languages remains a big challenge in the context of machine translation. In this paper, we propose to de-couple machine translation from morphology generation in order to better deal with the problem. We investigate the morphology simplification with a reasonable trade-off between expected gain…
This note explains how to transform Heegaard diagrams into framed link diagrams.
problem No specific problem stated; transformation of diagrams is the focus.
method Explains a procedure to transform Heegaard diagrams into framed link diagrams.
result Demonstrates a method to transform Heegaard diagrams into framed link diagrams.
We present a simplification of Neumann's formula for the universal Cheeger-Chern-Simons class of the second Chern polynomial. Our approach is completely algebraic, and the final formula can be applied directly on a homology class in the bar complex.
New minimal link diagrams found, including torus links and homogeneous ones.
problem Finding minimal link diagrams with new classes.
method Morton-Franks-Williams inequality approach.
result New classes of minimal link diagrams, including previously unproven ones.
This is a survey of some of the recent developments in the theory of complex Monge-Ampere equations. The topics discussed include refinements and simplifications of classical a priori estimates, methods from pluripotential theory, variational methods for big cohomology classes, semiclassical constructions of solutions …
Algorithm converts Kirby diagrams to trisection diagrams for 4-manifolds.
problem Creating efficient trisection diagrams for 4-manifolds.
method Algorithm converting Kirby diagrams to trisection diagrams.
result Provides examples of trisection diagrams for 4-manifolds.