A symmetric quandle is a quandle with a good involution. For a knot in \$R^3\$, a knotted surface in \$R^4\$ or an \$n\$-manifold knot in \$R^{n+2}\$, the knot symmetric quandle is defined. We introduce the notion of a symmetric quandle presentation, and show how to get a presentation of a knot symmetric quandle from a…
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48 results for “knot symmetric quandle”
Characterizes knot groups and symmetric quandles of surface-links.
problem Characterize knot groups and symmetric quandles of surface-links.
method Used plat presentations for surface-links and closed 2-dimensional braids.
result Generalized results to include non-orientable surface-links and showed that dihedral quandles can be realized as symmetric quandles of surface-links.
Computation of the knot symmetric quandle and its application to the plat index of surface-linksmath.GT
Paper computes knot symmetric quandle for surface-links and finds infinitely many distinct surface-knots.
problem Computing knot symmetric quandle for surface-links.
method Using plat form presentations, the paper computes the knot symmetric quandle for surface-links.
result Infinitely many distinct surface-knots of genus g with plat indices m.
The paper defines conditions for good involutions in generalized Alexander quandles.
problem Determining conditions for good involutions in generalized Alexander quandles.
method Analyzing the structure of generalized Alexander quandles and their involutions.
result Classification of all good involutions in connected generalized Alexander quandles.
New symmetric quandles constructed from group elements and subgroups.
problem Understanding the structure of symmetric quandles.
method Constructing symmetric quandles from specific group elements and subgroups.
result Every symmetric quandle is isomorphic to the disjoint union of constructed quandles.
The paper studies quandles from symmetric group automorphisms and finds a correspondence with conjugacy classes.
problem Investigating quandles from automorphisms of symmetric groups.
method Developed quandle invariants and proved a one-to-one correspondence between quandles and conjugacy classes.
result There is a one-to-one correspondence between generalized Alexander quandles arising from symmetric groups $\Sf_n$ and the conjugacy classes of $\Sf_n$ for with .
Paper introduces tensor product of quandles for knot classification.
problem Classifying knot invariants of surface-links with 1-handles.
method Introduces tensor product of quandles and applies to surface-links.
result Tensor product of knot quandles/classical quandles can classify surface-link invariants.
Quandles with involutions that satisfy certain conditions, called good involutions, can be used to color non-orientable surface-knots. We use subgroups of signed permutation matrices to construct non-trivial good involutions on extensions of odd order dihedral quandles. For the smallest example of order 6 that is an ex…
Classifies good involutions in conjugation subquandles and racks.
problem Classifying quandles with good involutions for applications in surface-knot theory.
method Study of subquandles of conjugation quandles, including core quandles; analysis of good involutions of faithful racks.
result Sharp bounds on the number of good involutions of racks in these families.
A quandle is a self-distributive algebraic structure that appears in quasi-group and knot theories. For each abelian group A and c \in A we define a quandle G(A, c) on \Z_3 \times A. These quandles are generalizations of a class of non-medial Latin quandles defined by V. M. Galkin so we call them Galkin quandles. Each …
Symmetric quandles provide new insights into link colorings.
problem Understanding link colorings using quandles.
method Construction of symmetric quandles and isomorphism of their cohomology groups.
result Homology groups of quandles are isomorphic to those of their symmetric doubles.
This paper computes the second quandle homology group of knot n-quandles.
problem Characterizing knots using quandle homology groups.
method Computation of the second quandle homology group for knot n-quandles.
result The second quandle homology group of knot n-quandles provides more information than the knot quandle's homology group.
New knot quandles distinguish ribbon knots with isomorphic groups.
problem Distinguishing knots with isomorphic fundamental groups.
method Examined knot quandles of Suciu's ribbon knots and computed their types.
result Knot quandles of Suciu's ribbon knots are mutually non-isomorphic.
Study explores associated groups of symmetric quandles and their properties.
problem Understanding the structure and relationships of symmetric quandles and their associated groups.
method Group-theoretic analysis and characterization of associated groups.
result Characterization and properties of associated groups of symmetric quandles.
The paper reinterprets knot group invariants using affine transformations.
problem Alexander invariants of knots and their geometric interpretation.
method Representation varieties of knot groups into .
result Alexander polynomial as the singular locus of a coherent sheaf.
This paper constructs quandles with abelian inner automorphism groups from graphs, proving their homogeneity.
problem Finding quandles with specific automorphism properties.
method Starting from simple graphs, the paper constructs quandles with abelian inner automorphism groups and proves their homogeneity.
result Homogeneous quandles with abelian inner automorphism groups are constructed from vertex-transitive graphs.
New 2-knots found with same knot group but different quandles.
problem Identifying 2-knots with identical knot groups but distinct quandles.
method Analyzing knot quandles of twist spins.
result First example of 2-knots with same knot group but different quandles.
New (co)homology theory for symmetric quandles developed.
problem Developing strong invariants for symmetric quandles.
method Introducing symmetric quandle modules and Beck modules, extending module theory, and constructing generalized (co)homology.
result Established an explicit isomorphism between symmetric quandle cohomology and group cohomology.
Study on quandle coloring quivers for (p, 2)-torus knots and links.
problem Understanding quandle colorings of (p, 2)-torus knots and links.
method Introduced quandle coloring quivers and studied them for dihedral quandles.
result Characterized quandle coloring quivers for (p, 2)-torus knots and links.
New method uses quandle rings to distinguish knots and their mirrors.
problem Distinguishing knots and their mirror images.
method Combining idempotents in quandle rings with state sum invariants.
result Improved knot and mirror image distinction using fewer quandles.
New knot quandle structure for twist-spun trefoils discovered.
problem Understanding symmetries of knot-spun structures.
method Defined Schläfli quandles and studied their relationship with knot quandles.
result The knot quandle of the twist-spun trefoil is a central extension of a Schläfli quandle.
Study satellite knots and their quandles related to incompressible tori.
problem Understanding quandles of satellite knots and their components.
method Algebraic approach to augmented fundamental quandles, presentations of fundamental quandles, and analysis of Alexander modules.
result Relationships between satellite knots, companion and pattern knots, and their fundamental quandles.
We show that the structure of a fibered knot, as a fiber bundle, is reflected in its knot quandle. As an application, we discuss finiteness and equivalence of knot quandles of concrete fibered 2-knots.
New exact sequence links cohomology, automorphisms, and extensions of symmetric quandles.
problem Understanding the structure of extensions and automorphisms in symmetric quandles.
method Derived a four-term exact sequence relating 1-cocycles, second cohomology, and automorphisms.
result Obstruction to automorphisms lies in the second cohomology of symmetric quandles.
We construct elements of the third quandle homology groups of knot quandles, which are called the shadow fundamental classes. They play the same roles for the shadow quandle cocycle invariants of knots as the fundamental classes of knot quandles does for the quandle cocycle invariants. As an application of the shadow f…
For an oriented knot , we construct a functor from the category of pointed quandles to the category of quandles in three different ways. We also extend the quandle cocycle invariants of knots by using these quandle-valued invariant of knots, and study their properties.
In this note, residual finiteness of quandles is defined and investigated. It is proved that free quandles and knot quandles of tame knots are residually finite and Hopfian. Residual finiteness of quandles arising from residually finite groups (conjugation, core and Alexander quandles) is established. Further, residual…
Quandle Module Quiversmath.GT
Enhances knot and link invariants using quandle modules.
problem Distinguishing knots and links using polynomial invariants.
method Integrates quandle modules into the quandle coloring quiver.
result The enhanced invariant distinguishes knots and links.
The study examines biorderability of prime knots up to eight crossings.
problem Determining biorderability of knot quandles for prime knots up to eight crossings.
method Analyzing knot quandles and generating linear orders that could be extended to biorders.
result Proves non-biorderability for specific knots and finds biorderable for others.
Paper provides criteria to detect non-admissible quandles via coloring.
problem Determining non-admissibility of quandles.
method Using colorings of (1, 1)-tangles to detect non-admissibility.
result Constructed numerous non-admissible quandles.
A quandle coloring obstruction prevents a specific link from being ribbon concordant.
problem Obstructing a specific link from being ribbon concordant.
method Symmetric dihedral quandle coloring analysis.
result A symmetric dihedral quandle of order 4 cannot color a specific surface-link, obstructing ribbon concordance.
The homology and cohomology of quandles and racks are used in knot theory: given a finite quandle and a cocycle, we can construct a knot invariant. This is a quick introductory survey to the invariants of knots derived from quandles and racks.
The paper explores orderability in link quandles and provides various results.
problem Orderability of link quandles and its interconnections with link groups.
method Develops a general theory of orderability, proves specific cases, and explores interconnections.
result Link quandles of most non-trivial torus links are not right-orderable.
Study Alexander matrices for link quandles and their relation to knot invariants.
problem Understanding Alexander matrices for link quandles and their applications to knot invariants.
method Investigate -twisted Alexander matrices and their connection to quandle cocycle invariants.
result Show that -twisted Alexander invariants of knot quandles are stronger than those of knot groups.
Knot quandle decompositionsmath.GT
We show that the fundamental quandle defines a functor from the oriented tangle category to a suitably defined quandle category. Given a tangle decomposition of a link , the fundamental quandle of may be obtained from the fundamental quandles of tangles. We apply this result to derive a presentation of the funda…
Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle invariants of composite knots. We investigate t…
The paper defines and calculates Euler characteristics for quandles.
problem Defining and calculating Euler characteristics for quandles.
method Definition and calculation of Euler characteristics for quandles.
result The quandle Euler characteristic of a compact connected Riemannian symmetric space coincides with the topological Euler characteristic.
Algebraic homology and cohomology theories for quandles have been studied extensively in recent years. With a given quandle 2(3)-cocycle one can define a state-sum invariant for knotted curves(surfaces). In this paper we introduce another version of quandle (co)homology theory, say positive quandle (co)homology. Some p…
Quandles can be regarded as generalizations of symmetric spaces. In the study of symmetric spaces, the notion of flatness plays an important role. In this paper, we define the notion of flat quandles, by referring to the theory of Riemannian symmetric spaces, and classify flat connected finite quandles.
We prove that the fundamental quandle of the trefoil knot is isomorphic to the projective primitive subquandle of transvections of the symplectic space . The last quandle can be identified with the Dehn quandle of the torus and the cord quandle on a 2-sphere with four punctures. We also show that the fund…
The paper explores idempotents in quandle rings and their connections to quandle coverings.
problem Understanding idempotents in quandle rings and their relation to quandle coverings.
method Investigation of idempotents in quandle rings, proving properties of idempotents in free products and unions of quandles.
result Integral quandle rings of quandles of finite type that are non-trivial coverings over nice base quandles admit infinitely many non-trivial idempotents.
If a knot has the Alexander polynomial not equal to 1, then it is linear -colorable. By means of such a coloring, such a knot is given an upper bound for the minimal quandle order, i.e., the minimal order of a quandle with which the knot is quandle colorable. For twist knots, we study the minimal quandle orders in d…
Dehn quandles of groups and surfaces unify various quandle constructions.
problem Understanding and unifying various quandle constructions.
method Introducing Dehn quandles of groups and subsets, proving properties and embeddings.
result Dehn quandles embed naturally into their enveloping groups, and enveloping groups of certain quandles are the quandles themselves.
New proof shows knot quandles of ribbon knots are distinct.
problem Proving knot quandles of ribbon knots are non-isomorphic.
method Analyzing conjugacy classes of automorphisms of a free group.
result Knot quandles of ribbon knots are mutually non-isomorphic.
Defined by Joyce and Matveev, the fundamental quandle is a complete invariant of oriented classical knots. We consider invariants of knots defined from quotients of the fundamental quandle. In particular, we introduce the fundamental Latin Alexander quandle of a knot and consider its Gröbner basis-valued invariants, wh…
Quandle Action Quiversmath.GT
New polynomial invariants from quandle action quivers.
problem Classical and virtual knot and link invariants.
method Categorification of quandle counting invariant using quandle action quivers.
result Quandle action polynomials as decategorifications.
We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of…
Note on spherical quandlesmath.GT
The paper connects knot representations and spherical quandle colorings.
problem Defining the Casson-Lin invariant for knots.
method One-to-one correspondence between -representations and spherical quandle colorings.
result Geometric interpretation of the trace-free condition for the Casson-Lin invariant.