Simplified proof for lattice link projections.
problem Necessary and sufficient condition for lattice link projections.
method Shorter and simpler proof of existing result.
result Simplified proof for lattice link projections.
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48 results for “lattice links”
New lattice stick knot condition identified.
problem Determining if 2D lattice knots project to 3D lattice sticks.
method Provided a necessary and sufficient condition.
result Identified a condition for 2D lattice knots to project to 3D lattice sticks.
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.
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.
This paper finds upper bounds for lattice stick numbers of rational links with specific stick configurations.
problem Finding upper bounds for the lattice stick number of rational links with exactly 4 z-sticks.
method Using 2-circuit presentations, the paper constructs lattice stick numbers with exactly 4 z-sticks and derives upper bounds.
result Upper bounds for the lattice stick number of rational links with exactly 4 z-sticks are derived.
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…
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.
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…
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.
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.
Formula for 2-bridge link Alexander polynomials via integer walks.
problem Computing Alexander polynomials of 2-bridge links.
method Extending Hartley's and Minkus' formula to 2-variable polynomials via integer lattice walks.
result A formula corresponding to a walk on the 2D integer lattice.
String theory connects lattice models, links, and geometric Langlands.
problem Connecting lattice models, links, and geometric Langlands.
method T-duality and worldvolume theories in string theory.
result Unified understanding of various mathematical concepts.
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.
New lattices are linked to higher hypergeometric functions.
problem Understanding non-arithmetic lattices in PU(2,1).
method Showed all known non-arithmetic lattices are monodromy groups of higher hypergeometric functions.
result Non-arithmetic lattices in PU(2,1) are linked to higher hypergeometric functions.
Researchers prove conjecture about contractible subcomplexes in noncrossing partition link.
problem Understanding contractibility of subcomplexes in the noncrossing partition link.
method Combining contractibility of flag complexes' stars with noncrossing hypertrees theory.
result Proved conjecture about contractible subcomplexes in the noncrossing partition link.
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…
Legendrian invariant studied in knot lattice homology.
problem Legendrian invariant in knot lattice homology.
method Defined Alexander grading and used filtered chain homotopy.
result Alexander grading of Legendrian invariant is invariant under blow-ups.
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.
Minimal lattice generating set size linked to group co-volume.
problem Determining minimal generating set size for lattices in Lie groups.
method Relies on semi-simple and solvable cases, extends to amenable groups.
result Bounding minimal generating set size by co-volume of projection.
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 lifts torus link homology.
problem Computing the triply-graded Khovanov-Rozansky homology of torus links.
method Defined a symmetric function and showed it satisfies a recursion for torus link homology.
result Triply-graded Khovanov-Rozansky homology of torus links is a specialization of .
Study on contact structures of surface singularity links using Heegaard Floer homology.
problem Understanding contact structures on singularity links using Heegaard Floer homology.
method Interplay between Heegaard Floer homology and Némethi's lattice cohomology.
result Heegaard Floer invariant cannot lie in the image of the U-action for canonical contact structures on singularity links.
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 , up to orientation-reversal.
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 i…
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.
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…
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.
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.
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…
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.
The article classifies cubiquitous sublattices and applies them to branched covers.
problem Understanding cubiquitous sublattices as obstructions to rational homology 4-balls.
method Developed a geometric Wu obstruction to classify cubiquitous sublattices and applied it to branched covers.
result Completely classified which sublattices with orthogonal bases are cubiquitous.
We prove that the signature of an even, symmetric form on a finite rank integral lattice, has signature divisible by 8, provided its associated linking form vanishes in the Witt group of linking forms. Our result generalizes the well know fact that an even, unimodular form has signature divisible by 8. We give applicat…
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.
Abstract framework for no-arbitrage concepts in topological vector lattices.
problem Generalization of no-arbitrage concepts in topological vector lattices.
method Imposing a structural condition on trading strategies and deriving abstract FTAP.
result NUPBR, NAA, and NA may not be equivalent in general setting.
The lattice cohomology of a plumbed 3--manifold associated with a connected negative definite plumbing graph is an important tool in the study of topological properties of , and in the comparison of the topological properties with analytic ones when is realized as complex analytic singularity link. By defini…
A combinatorial description of the centralizer algebras connected to the Links-Gould Invariantmath.QA
The paper proves a conjecture about the dimensions of centralizer algebras related to quantum super-algebras.
problem Proving a conjecture about the dimensions of centralizer algebras.
method Using combinatorial paths in a planar lattice, the authors describe the intertwiner spaces and provide a matrix unit basis.
result The conjecture about the dimensions of centralizer algebras is proven.
Explains a 2D color exchange invariant correspondence to 3D linking numbers.
problem Understanding color exchange invariants in 2D dynamics and their 3D geometric interpretation.
method Visualizes invariants as linking of lines on a special surface with Arf-Kervaire invariant one, and interprets it as an obstruction to continuous transformation.
result Interprets a 2D color exchange invariant as a 3D linking number, providing a topological explanation.
Study shows how volume growth affects homological torsion in 3-manifolds.
problem Understanding homological torsion growth in 3-manifolds.
method Examined abelian covers of alternating links and computed homological torsion growth explicitly.
result Explicit computation of homological torsion growth in terms of volume growth.
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.
Let l be a link of d components. For every finite-index lattice in Z^d there is an associated finite abelian cover of S^3 branched over l. We show that the order of the torsion subgroup of the first homology of these covers has exponential growth rate equal to the logarithmic Mahler measure of the Alexander polynomial …
Identifies arithmetic hyperbolic lattices generated by elements of order 4 and p, finding constraints and properties.
problem Identifying arithmetic hyperbolic lattices in 3D hyperbolic space.
method Analyzing presentations and properties of lattices generated by specific orders of elements.
result Necessary orders of p are {2, 3, 4, 5, 6, ∞}, and constraints on invariant trace fields and orbifolds.
The Whitehead link exterior lacks most Euler class taut foliations.
problem Existence of co-orientable taut foliations with specific Euler classes.
method Combining foliation theory techniques and topological obstructions.
result Most lattice points in the dual Thurston ball cannot be Euler classes of co-orientable taut foliations.
Simplicial versions of topological abelian gauge theories are constructed which reproduce the continuum expressions for the partition function and Wilson expectation value of linked loops, expressible in terms of R-torsion and linking numbers respectively. The new feature which makes this possible is the introduction o…
New unitary representations for Thompson's group V and virtual links.
problem Constructing unitary representations for Thompson's group V and virtual links.
method Defining a surjection from V to virtual equivalence classes of CC links, constructing unitary representations from kei and operator quandle colorings.
result Realizes all oriented almost classical virtual links.