We define and prove properties of link lattice complexes for plumbed links.
problem Understanding the homology and formality of plumbed L-space links.
method Define and analyze link lattice complexes, proving homotopy equivalence and formality properties.
result Link lattice complexes are homotopy equivalent to link Floer complexes for plumbed links.
Knot lattice homology invariant of smooth knot type in rational homology spheres.
problem Invariance of knot lattice homology in rational homology spheres.
method Proving knot lattice homology invariant through doubly-filtered homotopy type.
result Knot lattice homology invariant of smooth knot type in rational homology spheres.
There are two natural simplicial complexes associated to the noncrossing partition lattice: the order complex of the full lattice and the order complex of the lattice with its bounding elements removed. The latter is a complex that we call the noncrossing partition link because it is the link of an edge in the former. …
Study on entanglement complexity of confined ring polymers in lattice tubes.
problem Understanding the entanglement complexity of confined ring polymers in lattice tubes.
method Applied knot theory to extend and prove results about the complexity of 2SAPs.
result Proved that all but exponentially few size m 2SAPs have F complexity that grows at least linearly in m as m approaches infinity.
We give a simple example showing that a knot or link diagram that lies in the Z2 lattice is not necessarily the projection of a lattice stick knot or link in the Z3 lattice, and we give a necessary and sufficient condition for when a knot or link diagram that lies in the Z2 lat…
The paper studies alternating links in thickened surfaces using flow lattices and disc mutations.
problem Understanding alternating links in thickened surfaces and their invariants.
method Using integer flows on Tait graphs and disc mutations, the paper proves invariants and compares link properties.
result Found alternating knots with isometric flow lattices but different linking forms.
Knots and links have been considered to be useful models for structural analysis of molecular chains such as DNA and proteins. One quantity that we are interested on molecular links is the minimum number of monomers necessary to realize them. In this paper we consider every link in the cubic lattice. Lattice stick numb…
We give a shorter and simpler proof of the result of [2], which gives a necessary and sufficient condition for when a lattice diagram is the projection of a lattice link.
The cubic lattice stick index of a knot type is the least number of sticks necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p,p+1)-torus knots, and show how composing and taking satellites can be used to obtain t…
The paper studies gradings on nilpotent Lie algebras linked to smooth algebraic varieties.
problem Understanding gradings on nilpotent Lie algebras associated with algebraic varieties.
method Analyzing lattice structures in nilpotent Lie groups and their fundamental groups.
result Conditions for a lattice to be the fundamental group of a smooth complex algebraic variety.
The lattice stick number sL(L) of a link L is defined to be the minimal number of straight line segments required to construct a stick presentation of L in the cubic lattice. Hong, No and Oh found a general upper bound sL(K)≤3c(K)+2. A rational link can be represented by a lattice presentation with exa…
Alternating links bound rational homology balls if their chessboard lattice is cubiquitous.
problem When do alternating links bound rational homology balls?
method Heegaard Floer homology and flows on planar graphs.
result The normalized determinant of the link's chessboard lattice is a necessary and sufficient condition for the link to bound a rational homology ball.
Link condition for simplicial complexes to be CUB spaces.
problem Understanding when a simplicial complex is a CUB space.
method Establishing a link condition based on local lattice properties.
result The link condition generalizes Gromov's link condition for cube complexes.
Determines surgeries on chain links bounding rational homology balls using lattice-theoretic methods.
problem Integral surgeries on chain links bounding rational homology balls.
method Lattice-theoretic cubiquity obstruction and practical computation methods.
result Proves slice-ribbon conjecture for quasi-alternating 3-braid links, extending previous results.
The d-invariant of an integral, positive definite lattice L records the minimal norm of a characteristic covector in each equivalence class mod 2L. We prove that the 2-isomorphism type of a connected graph is determined by the d-invariant of its lattice of integral cuts (or flows). As an application, we prove that a re…
Analytic lattice cohomology defined for isolated singularities, linking to Heegaard Floer cohomology.
problem Defining and analyzing analytic lattice cohomology for isolated singularities.
method Using a good resolution of the singularity, proving independence of resolution choice, and relating to Hodge spectral numbers.
result Independence of analytic lattice cohomology from the choice of resolution and connection to Hodge spectral numbers.
Knots are commonly found in molecular chains such as DNA and proteins, and they have been considered to be useful models for structural analysis of these molecules. One interested quantity is the minimum number of monomers necessary to realize a molecular knot. The minimum lattice length $\mbox{Len}(K)$ of a knot K i…
Study links weaving knots with polynomial coefficients and lattice numbers.
problem Understanding polynomial coefficients of weaving knots and their lattice counterparts.
method Established relationships between Jones and Chebyshev polynomials, and derived explicit formulas for Alexander polynomials.
result Proved coefficients of Jones polynomial are Whitney numbers of Lucas lattices and satisfied Fox's trapezoidal conjecture.
Sharp bounds for spanning tree entropy in planar lattices.
problem Estimating spanning tree entropy in planar lattice graphs.
method Using hyperbolic geometry and polyhedra volumes.
result Proved bounds are easy to compute and provide excellent estimates.
Using the link surgery formula for Heegaard Floer homology we find a spectral sequence from the lattice homology of a plumbing tree to the Heegaard Floer homology of the corresponding 3-manifold. This spectral sequence shows that for graphs with at most two "bad" vertices, the lattice homology is isomorphic to the Heeg…
We compute the Heegaard-Floer link homology of algebraic links in terms of the multivariate Hilbert function of the corresponding plane curve singularities. The main result of the paper identifies four homologies: (a) the Heegaard-Floer link homology of the local embedded link of the germ, (b) the lattice homology asso…
New measures and tests for high-order interactions in complex data.
problem Challenges in capturing high-order interactions in multivariate data.
method Hierarchy of d-order interaction measures and kernel-based tests. result Established statistical significance of high-order interactions.
The paper finds incommensurable lattices in complex models of Baumslag-Solitar groups.
problem Locally finite 2-complexes and their automorphism groups contain incommensurable lattices.
method Constructing lattices in combinatorial models of Baumslag-Solitar groups and analyzing their properties.
result The constructed lattices are incommensurable and have specific properties like isomorphic Cayley graphs.
The lattice cohomology of a plumbed 3--manifold M associated with a connected negative definite plumbing graph is an important tool in the study of topological properties of M, and in the comparison of the topological properties with analytic ones when M is realized as complex analytic singularity link. By defini…
Proof of Knot Entropy Conjecture for tube lattice polygons.
problem Proving exponential growth rate of knot polygons equals unknot polygons.
method Upper and lower bounds on polygon counts, braid insertions, and pattern theorems.
result Established the Knot Entropy Conjecture for tube lattice polygons.
Symmetric function LM,N lifts torus link homology.
problem Computing the triply-graded Khovanov-Rozansky homology of torus links.
method Defined a symmetric function LM,N and showed it satisfies a recursion for torus link homology. result Triply-graded Khovanov-Rozansky homology of torus links is a specialization of LM,N. Hexagonal diagrams link complex curves in CP2 to minimal genus surfaces.
problem Understanding the relationship between complex curves and surfaces in CP2. method Hexagonal lattice diagrams and trisection of CP2. result Positive genus surfaces in CP2 are isotopic to complex curves if they admit hexagonal lattice diagrams. Study complex hyperbolic lattices and their relation to strict hyperbolization.
problem Understanding the relationship between complex hyperbolic lattices and strict hyperbolization.
method Analyzing the fundamental groups of complex hyperbolic manifolds and spaces arising from strict hyperbolization.
result Uniform lattices in PU(n,1) cannot be fundamental groups of Charney-Davis strict hyperbolizations when n ≥ 2.
The study connects lattices, Garside structures, and weakly modular graphs.
problem Exploring combinatorial non-positive curvature in various simplicial complexes.
method Analyzing lattices with Z-actions and their quotients. result Lattices and their quotients give rise to weakly modular graphs.
Researchers reinterpret complex hyperbolic orbifolds using line arrangements.
problem Understanding complex hyperbolic orbifolds and their representations.
method Using line arrangements and branched covers over blow-ups of projective 2-space.
result New representations of 3-manifolds and additional Deligne-Mostow lattices identified.
For any negative definite plumbed 3-manifold M we construct from its plumbed graph a graded Z[U]-module. This, for rational homology spheres, conjecturally equals the Heegaard-Floer homology of Ozsvath and Szabo, but it has even more structure. If M is a complex singularity link then the normalized Euler-characteristic…
Researchers find optimal configurations of complex knots and links.
problem Finding the most efficient configurations of complex knots and links.
method Minimizing Möbius and Minimum Distance energies by describing them with a small number of free parameters.
result Optimal geometries for Hopf links, Borromean rings, and chain links are found.
Study of subgroups in complex hyperbolic lattice triangle groups.
problem Characterizing subgroups of finite index in complex hyperbolic lattice triangle groups.
method Explicit construction and analysis of subgroups, examination of their properties.
result Identification of neat subgroups, subgroups with positive first Betti number, and homomorphisms onto non-Abelian free groups.
Study complex hyperbolic lattices and their subgroups, proving new finiteness properties.
problem Characterize subgroups of complex hyperbolic lattices.
method Analyzing homomorphisms and using arithmetic lattice properties.
result Deep subgroups of complex hyperbolic lattices admit homomorphisms to Z with specific kernel types.
Determines conditions for ribbon cobordisms between lens spaces.
problem Conditions for ribbon rational homology cobordisms between lens spaces.
method Analyzes ribbon cobordisms and uses properties of lens spaces and linear lattices.
result If a lens space admits a ribbon rational homology cobordism to a different lens space, it must be homeomorphic to L(n,1), up to orientation-reversal. Unified framework for complex financial networks using lattice theory.
problem Complex financial networks with multiple currencies and dependencies.
method Recast classical financial clearing model into lattice liability networks.
result Lattice-valued clearing sections form a complete lattice, enabling tractable analysis.
Study C-Fuchsian subgroups of non-arithmetic lattices.
problem Understand structure and fundamental domains of C-Fuchsian subgroups. method General procedure to analyze structure and show fundamental domains lie on a complex geodesic.
result Fundamental domains of C-Fuchsian subgroups lie on a complex geodesic homeomorphic to the unit disk. This paper introduces lattice representations for efficient discrete learning.
problem Efficient learning of discrete representations in Euclidean space.
method Lattice quantization and novel algorithms for efficient learning.
result New mathematical result linking training and inference expressions.
A formula for the Alexander polynomial of a 2-bridge knot or link given by Hartley and also by Minkus has a beautiful interpretation as a walk on the integers. We extend this to the 2-variable Alexander polynomial of a 2-bridge link, obtaining a formula that corresponds to a walk on the 2-dimensional integer lattice.
Researchers calculate complexity of billiard paths in regular polygons.
problem Calculating the complexity of billiard paths in regular polygons.
method Counting saddle connections on lattice surfaces, focusing on combinatorial length.
result They answered a question about billiard language complexity in regular polygons.
The paper classifies groups containing incommensurable lattices in Baumslag-Solitar complexes.
problem Classifying groups containing incommensurable lattices in Baumslag-Solitar complexes.
method Analyzing combinatorial automorphisms and properties of cell complexes.
result Conditions for the existence of incommensurable torsion-free lattices in Aut(Xm,n). Finite presentations for skein algebras linked to gauge field theory.
problem Understanding finite presentations for skein algebras and their relationship to gauge field theory.
method Provided finite presentations and deduced properties of stated skein algebras.
result Stated skein algebras are Koszul and isomorphic to quantum moduli algebras in gauge field theory.
By applying Seifert's algorithm to a special alternating diagram of a link L, one obtains a Seifert surface F of L. We show that the support of the sutured Floer homology of the sutured manifold complementary to F is affine isomorphic to the set of lattice points given as hypertrees in a certain hypergraph that is natu…
This paper studies posets associated with link diagrams and their algebraic properties.
problem Understanding the algebraic structure of posets derived from link diagrams.
method Associaed posets with link diagrams, proved distributivity, and described join irreducibles.
result Posets of Kauffman states are distributive lattices and isomorphic to coefficient quiver posets.
Let I(p,v) be Bourdon's building, the unique simply-connected 2-complex such that all 2-cells are regular right-angled hyperbolic p-gons and the link at each vertex is the complete bipartite graph K(v,v). We investigate and mostly determine the set of triples (p,v,g) for which there exists a uniform lattice Γ in Aut(I(…
Study stabilizers of complex hyperbolic triangle groups, finding generators and signatures.
problem Understanding the stabilizers of complex hyperbolic triangle groups.
method Explicit generators and signatures of stabilizers computed for each group orbit of mirrors.
result Explicit generators and signatures of stabilizers for some triangle groups.
The paper confirms a conjecture linking link bipyramid volume and Mahler measure.
problem Link bipyramid volume and Mahler measure relationship for alternating links.
method Using isoradial graphs and spanning trees on lattices, the authors confirm the conjecture for two examples and calculate five more.
result The conjecture is confirmed for specific examples of alternating links.
We prove there are exactly 16 arithmetic lattices of hyperbolic 3-space which are generated by two elements of finite orders p and q with p,q at least six. We also verify a conjecture of H.M. Hilden, M.T. Lozano, and J.M. Montesinos concerning the orders of the singular sets of arithmetic orbifold Dehn surgeries on two…