Algorithm calculates Jones polynomial from Goeritz matrix.
problem Calculating Jones polynomial from link diagrams.
method Explicit algorithm using Goeritz matrices.
result Jones polynomial can be recovered from orientable checkerboard surfaces.
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48 results for “Goeritz forms”
A new method calculates HOMFLY-PT polynomials for bipartite links.
problem Computing HOMFLY-PT polynomials for bipartite links efficiently.
method Generalizes Goeritz matrix method for bipartite links.
result Reduces HOMFLY-PT polynomial calculation to matrix algebra.
New criteria for Heegaard splittings ensure strong irreducibility and finite Goeritz groups.
problem Determining strong irreducibility and finite Goeritz groups of Heegaard splittings.
method Two diagrammatic criteria for Heegaard splittings, accepting arbitrary disk systems.
result Criteria ensure strong irreducibility and finite Goeritz groups for Heegaard splittings.
The paper constructs Goeritz matrices from Dehn colorings.
problem Constructing Goeritz matrices from Dehn colorings.
method Purely algebraic construction of Goeritz matrices from Dehn coloring matrices for prime knot diagrams.
result A new method to construct Goeritz matrices from Dehn colorings.
Link colorings linked to Goeritz matrix.
problem Understanding link colorings and their relation to the Goeritz matrix.
method Exploring the relationship between link colorings and the Goeritz matrix.
result Established a connection between link colorings and the Goeritz matrix.
Genus 2 Goeritz Equivalence in math.GT
Study on Goeritz equivalence in genus 2 Heegaard splitting of .
problem Understanding Goeritz equivalence of curves in genus 2 Heegaard splitting of .
method Introduce Goeritz equivalence of curves, present algebraic obstructions, and provide examples.
result Algebraic obstructions to Goeritz equivalence of simple closed curves are computed and demonstrated.
Study on 3-sphere Goeritz group's twisted first homology group.
problem Determining the twisted first homology group of a specific Goeritz group.
method Defined genus-g Goeritz group, analyzed 3-sphere with genus-2 Heegaard splitting.
result Determined the twisted first homology group of the genus-2 Goeritz group.
Powell's conjecture about Goeritz groups for genus 3 splittings is confirmed.
problem Determining the minimum set of elements needed to generate the Goeritz group of Heegaard splittings.
method Proof for splittings of genus 3 and a framework for higher genus splittings.
result Powell's conjecture is correct for splittings of genus 3.
Sharp signature bound for positive four-braids derived from three-braids.
problem Optimizing the signature bound for positive four-braids.
method Combining bounds for positive three-braids with Gordon and Litherland's approach to signature via unoriented surfaces and Goeritz forms.
result Improved linear bounds for the signature in terms of the three-genus of closures and first Betti number.
Finite Goeritz groups for links with long bridge decompositions.
problem Characterizing Goeritz groups of links.
method Proving finite Goeritz groups for links with specific bridge decompositions.
result Finite Goeritz groups for links with distance at least 6.
Link equivalence determined by Goeritz matrices.
problem Determining link equivalence through Goeritz matrices.
method Extending Greene and Lipson's theorems, proving equivalence class determination.
result Link equivalence determined by Goeritz matrices.
Goeritz and Seifert matrices derived from Dehn presentations.
problem Computing Goeritz and Seifert matrices for links.
method Using Fox's free differential calculus on modified Dehn presentations.
result Goeritz and Seifert matrices can be derived from Dehn presentations.
The paper studies Goeritz equivalence in lens spaces, describing actions and obstructions.
problem Understanding Goeritz equivalence in lens spaces for genus two Heegaard splittings.
method Describes the Goeritz group action on the homology of the Heegaard surface and provides obstructions.
result Homology and homotopy obstructions for Goeritz equivalence of curves in the Heegaard surface.
Researchers compute Goeritz groups for all (1,1)-link decompositions.
problem Computing Goeritz groups for all (1,1)-link decompositions.
method Analyzing surface decompositions and isotopy classes of homeomorphisms.
result Computed Goeritz groups for all (1,1)-link decompositions.
Generating the Goeritz group of math.GT
New generators prove sufficiency for Goeritz group of 3-sphere.
problem Proving sufficiency of specific generators for Goeritz group.
method Expanding Powell's proposed generators to include all eyeglass twists and topological conjugates.
result Natural expansion of Powell's generators suffices to generate Goeritz group.
Study on Goeritz groups from twisted book decompositions with new examples.
problem Understanding Goeritz groups in Heegaard splittings induced from twisted book decompositions.
method Explicit computation of mapping class groups from Heegaard splittings of distance 2.
result Existence of Heegaard splittings with infinite-order mapping class groups not induced from open book decompositions.
Study Goeritz groups of link decompositions, focusing on their asymptotic behavior.
problem Understanding the asymptotic behavior of Goeritz groups for link decompositions.
method Defined Goeritz groups for link decompositions, analyzed their properties, and discussed their asymptotic behavior.
result Discussed the asymptotic behavior of minimal pseudo-Anosov entropies and related it to Goeritz groups of Heegaard splittings.
New subgroup behavior in genus-2 mapping class group identified.
problem Understanding subgroups in genus-2 mapping class group.
method Analyzing purely pseudo-Anosov subgroups as convex cocompact.
result Finitely-generated, purely pseudo-Anosov subgroups are convex cocompact.
We obtain the full list of Goeritz invariants of all torus knots and links.
The paper discusses knot colorings and their invariants using Goeritz matrices.
problem Distinguishing knots using coloring methods.
method Elementary approach to equivalence between coloring and Goeritz matrices.
result Computing knot determinant and nullity of pretzel knots.
This paper studies a subgroup of the Goeritz group related to Heegaard splittings induced by openbook decompositions.
problem Understanding the subgroup of the Goeritz group associated with Heegaard splittings from openbook decompositions.
method Analyzes the mapping class group of a 3-manifold, focusing on elements that preserve the binding and commute with the monodromy.
result Characterizes the Goeritz group subgroup as a quotient of specific mapping class groups and provides a criterion for certain elements.
The genus two Goeritz group of math.GT
The genus-g Goeritz group is the group of isotopy classes of orientation-preserving homeomorphisms of a closed orientable 3-manifold that preserve a given genus-g Heegaard splitting of the manifold. In this work, we show that the genus-2 Goeritz group of is finitely presented, and give its explicit pre…
Given a genus- Heegaard splitting of a 3-manifold, the Goeritz group is defined to be the group of isotopy classes of orientation-preserving homeomorphisms of the manifold that preserve the splitting. In this work, we show that the Goeritz groups of genus-2 Heegaard splittings for lens spaces are finitely …
Authors prove a conjecture about the Goeritz group of 3-sphere Heegaard splittings.
problem The Goeritz group of a genus g Heegaard splitting of the 3-sphere.
method Proof relies on the topological minimality of Heegaard surfaces and their disk complexes.
result The Goeritz group is generated by four specific elements for g ≥ 3.
For all genus g, Powell's elements generate Goeritz groups trivially.
problem Tackles the stability of Powell's Conjecture on Goeritz groups of S^3.
method Shows that the natural function is trivial for each genus g.
result For each genus g, the natural function from G_g to G_{g+1}/P_{g+1} is trivial.
Finite set of Dehn twists describes genus-2 Goeritz group elements.
problem Describing elements of the genus-2 Goeritz group of .
method Finite generating set of Dehn twists, algorithm to describe elements, complexity measure on reducing spheres.
result Unique description of elements in terms of Dehn twists.
A specific set of 4g+1 elements is shown to generate the Goeritz group of the genus g+1 Heegaard splitting of a genus g handlebody. These generators are consistent with Powell's proposed generating set for the Goeritz group of the genus g+1 splitting of S^3. There are two proofs: one using purely classical techniques a…
Adds examples to Goeritz groups for a specific type of 3-manifold splitting.
problem Characterizing Goeritz groups for a particular class of 3-manifolds.
method Analyzes Heegaard splittings of genus two Seifert manifolds with specific properties.
result Identifies new examples of Goeritz groups for the specified 3-manifolds.
Genus 3 Heegaard groups of lens space connected sums are finitely generated.
problem Understanding the structure of mapping class groups of genus 3 Heegaard splittings.
method Proved finitely generated property through connected reducing sphere complexes.
result Mapping class groups are finitely generated and complexes are connected.
Gordon-Litherland pairing connects combinatorics and topology.
problem Unifying quadratic forms in link theory.
method Picture proof using Kirby diagrams.
result Their theorem has numerous applications in low-dimensional topology.
One of Powell's generators is not necessary.
problem Unresolved conjecture about generating Goeritz group.
method Short argument showing redundancy of one generator.
result One of Powell's generators is a consequence of others.
Knot Theory is currently a very broad field. Even a long survey can only cover a narrow area. Here we concentrate on the path from Goeritz matrices to quasi-alternating links. On the way, we often stray from the main road and tell related stories, especially if they allow as to place the main topic in a historical cont…
According to a formula by Gordon and Litherland, the signature of a knot K can be computed as the signature of a Goeritz matrix of K minus a suitable correction term, read off from the diagram of K. In this article, we consider the family of two bridge knots K(p/q) and compute the signature of their Goeritz matrices in…
Knots and 4-manifolds linked via matrix kinking.
problem Understanding equivalence of symmetric matrices and their implications.
method Isotopy and kinking moves on Goeritz matrices.
result Every nonsingular symmetric integer matrix is kink-equivalent to positive or negative-definite matrices.
New knot invariant bounds crossing number and homology.
problem Bounding knot invariants and homology groups.
method Region crossing changes, Goeritz matrices.
result Minimum homology generators < multi-region index.
Connected primitive disk complexes help understand lens spaces.
problem Understanding the structure of genus two Goeritz groups of lens spaces.
method Study of primitive disk complexes for genus two Heegaard splittings of lens spaces.
result Connected primitive disk complexes for lens spaces are characterized by specific congruences of and .
Study connects Powell Conjecture to reducing sphere complex's connectivity.
problem Prove Powell Conjecture for genus Heegaard surfaces.
method Analyze reducing sphere complex and its connectedness.
result Powell Conjecture true if and only if is connected.
We show that if a Heegaard splitting is obtained by gluing a splitting of Hempel distance at least 4 and the genus-1 splitting of , then the Goeritz group of the splitting is finitely generated. To show this, we first provide a sufficient condition for a full subcomplex of the arc complex for a compact …
Confirming the Powell Conjecture for genus-3 Heegaard splittings of the 3-sphere.
problem Proving the finitely generated nature of the Goeritz group for genus-3 Heegaard splittings of the 3-sphere.
method Establishing the connectivity of reducing sphere complexes for the genus-3 case.
result Confirmation of the Powell Conjecture for genus-3 Heegaard splittings of the 3-sphere.
New method shows how to move one Heegaard surface positioning to another.
problem Moving Heegaard surfaces in a specific way.
method Isotopies, pushing stabilizing pairs through, eyegelass twists.
result Shows how to move one Heegaard surface positioning to another.
Algorithm constructs reducing spheres for genus-2 Heegaard splitting of S^3.
problem Finite generation of Goeritz group .
method Algorithm to construct reducing spheres from a standard reducing sphere.
result Alternate proof of finite generation of .
For a Heegaard surface F in a closed orientable 3-manifold M, H(M,F) = Diff(M)/Diff(M,F) is the space of Heegaard surfaces equivalent to the Heegaard splitting (M,F). Its path components are the isotopy classes of Heegaard splittings equivalent to (M,F). We describe H(M,F) in terms of Diff(M) and the Goeritz group of (…
Invalidation of a key lemma leaves the Powell Conjecture unresolved.
problem Invalidation of Lemma 2.5 in the Powell Conjecture proof.
method Analysis of Lemma 2.5's validity in the context of the Powell Conjecture.
result Lemma 2.5 is invalid, leaving the Powell Conjecture unresolved.
2-width of 3-manifolds is bounded by 2 but embeddings can have infinite 2-width.
problem Defining and analyzing the 2-width of 3-manifolds and their embeddings.
method Generalizing width concept to 3-manifolds and embeddings, showing bounds and divergences.
result Embeddings of 3-manifolds can have arbitrarily large 2-width, challenging classical width concepts.
An updated proof of a 1933 theorem of Goeritz, exhibiting a finite set of generators for the group of automorphisms of the 3-sphere that preserve a genus two Heegaard splitting. The group is analyzed via its action on a certain connected 2-complex. (The analogous problem for higher genus Heegaard splittings appears to …
Machine learning maps knots to embeddings, revealing topological invariants.
problem Learning topological invariance in knot theory.
method Contrastive and generative machine learning techniques, auto-regressive decoder Transformer network.
result Neural networks can map different knots to the same point in an embedding vector space.
The paper connects knot invariants to Laplacian matrices.
problem Computing knot invariants for various link diagrams.
method Using Laplacian matrices of directed, edge-weighted graphs derived from link diagrams.
result Principal minors of Laplacian matrices yield Seifert, Alexander, Goeritz matrices.
Given a genus two Heegaard splitting for a non-prime 3-manifold, we define a special subcomplex of the disk complex for one of the handlebodies of the splitting, and then show that it is contractible. As applications, first we show that the complex of Haken spheres for the splitting is contractible, which refines the r…