A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Dehn quandles of groups and surfaces unify various quandle constructions.
The paper studies finiteness of canonical quotients in Dehn quandles of surfaces.
New quandles classify surfaces, with infinite cohomology.
In-degree quiver polynomials for surface-links computed.
Paper introduces tensor product of quandles for knot classification.
By using the cohomology theory of quandles, quandle cocycle invariants and shadow quandle cocycle invariants are defined for oriented links and surface-links via broken surface diagrams. By using symmetric quandles, symmetric quandle cocycle invariants are also defined for unoriented links and surface-links via broken …
A quandle coloring obstruction prevents a specific link from being ribbon concordant.
Paper computes knot symmetric quandle for surface-links and finds infinitely many distinct surface-knots.
Biquandles are generalizations of quandles. As well as quandles, biquandles give us many invariants for oriented classical/virtual/surface links. Some invariants derived from biquandles are known to be stronger than those from quandles for virtual links. However, we have not found an essentially refined invariant for c…
Characterizes knot groups and symmetric quandles of surface-links.
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…
Symmetric quandles provide new insights into link colorings.
Study of twisted Alexander matrices for certain quandles and their invariants.
We introduce the notion of a quandle with a good involution and its homology groups. Carter et al. defined quandle cocycle invariants for oriented links and oriented surface-links. By use of good involutions, quandle cocyle invariants can be defined for links and surface-links which are not necessarily oriented or orie…
New method for quandle presentations of surface knots in 4-manifolds.
The paper presents methods to write presentations for Dehn quandles.
We show that isotopy classes of simple closed curves in any oriented surface admit a quandle structure with operations induced by Dehn twists, the Dehn quandle of the surface. We further show that the monodromy of a Lefschetz fibration can be conveniently encoded as a quandle homomorphism from the knot quandle of the b…
We have a knot quandle and a fundamental class as invariants for a surface-knot. These invariants can be defined for a classical knot in a similar way, and it is known that the pair of them is a complete invariant for classical knots. In this paper, we compare a situation in surface-knot theory with that in classical k…
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…
New shifting chain map enhances quandle invariants for links.
The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a state-sum invariant for classical knots and knotted surfaces is developed via a cohomology theory of racks a…
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…
Maps quandle homology to relative group homology.
Classifies good involutions in conjugation subquandles and racks.
State-sum invariants for knotted curves and surfaces using quandle cohomology were introduced by Laurel Langford and the authors in math.GT/9903135 In this paper we present methods to compute the invariants and sample computations. Computer calculations of cohomological dimensions for some quandles are presented. For c…
Study fundamental quandle of ribbon concordances, proving homomorphisms.
New exact sequence links cohomology, automorphisms, and extensions of symmetric quandles.
The state-sum invariants for knots and knotted surfaces defined from quandle cocycles are described using the Kronecker product between cycles represented by colored knot diagrams and a cocycle of a finite quandle used to color the diagram. Such an interpretation is applied to evaluating the invariants. Algebraic inter…
New (co)homology theory for symmetric quandles developed.
Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Graña. We specialize that theory to the case when there is a group action on the coefficients. First, quandle modules are used to generalize Burau representations and Alexander module…
Homomorphisms on quandle cohomology groups that raise the dimensions by one are studied in relation to the cocycle state-sum invariants of knots and knotted surfaces. Skein relations are also studied.
This paper is a brief overview of some of our recent results in collaboration with other authors. The cocycle invariants of classical knots and knotted surfaces are summarized, and some applications are presented.
A quandle is a set that has a binary operation satisfying three conditions corresponding to the Reidemeister moves. Homology theories of quandles have been developed in a way similar to group homology, and have been applied to knots and knotted surfaces. In this paper, a homology theory is defined that unifies group an…
Paper classifies surface-links using charts with specific properties.
It is known that any surface knot can be transformed to an unknotted surface knot or a surface knot which has a diagram with no triple points by a finite number of 1-handle additions. The minimum number of such 1-handles is called the unknotting number or the triple point cancelling number, respectively. In this paper,…
The theory of quandle (co)homology and cocycle knot invariants is rapidly being developed. We begin with a summary of these recent advances. One such advance is the notion of a dynamical cocycle. We show how dynamical cocycles can be used to color knotted surfaces that are obtained from classical knots by twist-spinnin…
Study of quandle coloring quivers with dihedral quandles.
We introduce a new construction of a surface link in the 4-space. We construct a surface link as a branched covering over the standard torus, which we call a torus-covering link. We show that a certain torus-covering -link is equivalent to the split union of spun -links and turned spun -links. We show th…
The paper explores idempotents in quandle rings and their connections to quandle coverings.
Paper studies embedding conditions for homogeneous quandles.
A spatial surface is a compact surface embedded in the 3-sphere. In this paper, we provide several typical examples of spatial surfaces and construct a coloring invariant to distinguish them. The coloring is defined by using a multiple group rack, which is a rack version of a multiple conjugation quandle.
Hierarchical quandles extend diquandles and multi-quandles for link invariants.
The paper studies invariants of surfaces in the 3-sphere using handlebody-links.
New polynomial invariants from quandle action quivers.
This paper computes the second quandle homology group of knot n-quandles.
Quantum cocycle invariants derived from Yang-Baxter cohomology.
The paper defines conditions for good involutions in generalized Alexander quandles.
The aim of this paper is to propose a theory of derivations for quandles. Given a quandle admitting an action by a quandle , derivations from to are introduced as twisted analogues of quandle homomorphisms. It is shown that for each quandle there exists a unique -quandle (the deriv…