arXiv subjects
Publications and source records attributed to Louis H. Kauffman.
We define a Khovanov-Lipshitz-Sarkar stable homotopy type for the homotopical Khovanov homology of links in the thickened torus after the authors introduced that in the case of higher genus surfaces in the previous paper of this one.
Vortex knots have been seen decaying in many physical systems. Here we describe topologically protected vortex knots, which remain stable and undergo fusion and fission while conserving a topological invariant analogous to that of baryon number. While the host medium, a chiral nematic liquid crystal, exhibits intrinsic chirality, cores of the vortex lines are structurally achiral regions where twist cannot be defined. We refer to them as "dischiralation" vortex lines, in analogy to dislocations and disclinations in ordered media where, respectively, positional and orientational order is disrupted. Fusion and fission of these vortex knots, which we reversibly switch by electric pulses, vividly reveal the physical embodiments of knot theory's concepts like connected sums of knots. Our findings provide insights into related phenomena in fields ranging from cosmology to particle physics and can enable applications in electro-optics and photonics, where such fusion and fission processes can be used for controlling light.
Multi-virtual knot theory was introduced in $2024$ by the first author. In this paper, we initiate the study of algebraic invariants of multi-virtual links. After determining a generating set of (oriented) multi-virtual Reidemeister moves, we discuss the equivalence of multi-virtual link diagrams, particularly those that have the same virtual projections. We introduce operator quandles (that is, quandles with a list of pairwise commuting automorphisms) and construct an infinite family of connected operator quandles in which at least one third of right translations are distinct and pairwise commute. Using our set of generating moves, we establish the operator quandle coloring invariant and the operator quandle $2$-cocycle invariant for multi-virtual links, generalizing the well-known invariants for classical links. With these invariants at hand, we then classify certain small multi-virtual knots based on the existing tables of small virtual knots due to Bar-Natan and Green. Finally, to emphasize a key difference between virtual and multi-virtual knots, we construct an infinite family of pairwise nonequivalent multi-virtual knots, each with a single classical crossing. Many open problems are presented throughout the paper.
In particle phenomenology, preon models study compositional rules of standard model interactions. In spite of empirical success, mathematical underpinnings of preon models in terms of group representation theory have not been fully worked out. Here, we address this issue while clarifying the relation between different preon models. In particular, we focus on two prominent models: Bilson-Thompson's helon model, and Lambek's 4-vector model. We determine the mapping between helon model particle states and representation theory of Lie algebras. Braided ribbon diagrams of the former represent on-shell states of spinors of the Lorentz group. Braids correspond to chirality, and twists, to charges. We note that this model captures only the $SU(3)_c\times U(1)_{em}$ sector of the standard model. We then map the twists of helon diagrams to the weight polytope of $SU(3)_c \times U(1)_{em}$. The braid structure maps to chiral states of fermions. We also show that Lambek's 4-vector can be recovered from helon diagrams. Alongside, we introduce a new 5-vector representation derived from the weight lattice. This representation contains both, the correct interactions found in 4-vectors and the inclusion of chirality found in helons. Additionally, we demonstrate topological analogues of CPT transformations in helon diagrams. Interestingly, the braid diagrams of the helon model are the only ones that are self-consistent with CPT invariance. In contrast to field-theoretic approaches, the compositional character of preon models offers an analogous particle-centric perspective on fundamental interactions.
The mock Alexander polynomial is an extension of the classical Alexander polynomial, defined and studied for (virtual) knots and knotoids by the second and third authors. In this paper we consider the mock Alexander polynomial for generalizations of knotoids. We prove a conjecture on the mock Alexander polynomial for knotoids, which generalizes to uni-linkoids. Afterwards we give constructions for canonical invariants of linkoids derived from the mock Alexander polynomial, using the formalism of generalized knotoids due to Adams et al.
We define three different types of spanning surfaces for knots in thickened surfaces. We use these to introduce new Seifert matrices, Alexander-type polynomials, genera, and a signature invariant. One of these Alexander polynomials extends to virtual knots and can obstruct a virtual knot from being classical. Furthermore, it can distinguish a knot in a thickened surface from its mirror up to isotopy. We also propose several constructions of Heegaard Floer homology for knots in thickened surfaces, and give examples why they are not stabilization invariant. However, we can define Floer homology for virtual knots by taking a minimal genus representative. Finally, we use the Behrens-Golla $δ$-invariant to obstruct a knot from being a stabilization of another.
In this paper, we construct mock Alexander polynomials for starred links and linkoids in surfaces. These polynomials are defined as specific sums over states of link or linkoid diagrams that satisfy $f=n$, where $f$ denotes the number of regions and $n$ denotes the number of crossings of diagrams.
The entanglement of open curves in 3-space appears in many physical systems and affects their material properties and function. A new framework in knot theory was introduced recently, that enables to characterize the complexity of collections of open curves in 3-space using the theory of knotoids and linkoids, which are equivalence classes of diagrams with open arcs. In this paper, new invariants of linkoids are introduced via a surjective map between linkoids and virtual knots. This leads to a new collection of strong invariants of linkoids that are independent of any given virtual closure. This gives rise to a collection of novel measures of entanglement of open curves in 3-space, which are continuous functions of the curve coordinates and tend to their corresponding classical invariants when the endpoints of the curves tend to coincide.
The total face color polynomial is based upon the Poincaré polynomials of a family of filtered $n$-color homologies. It counts the number of $n$-face colorings of ribbon graphs for each positive integer $n$. As such, it may be seen as a successor of the Penrose polynomial, which at $n=3$ counts $3$-edge colorings (and consequently $4$-face colorings) of planar trivalent graphs. In this paper we describe a state sum formula for the polynomial. This formula unites two different perspectives about graph coloring: one based upon topological quantum field theory and the other on diagrammatic tensors.
We investigate operator algebraic origins of the classical Koopman-von Neumann wave function $ψ_{KvN}$ as well as the quantum mechanical one $ψ_{QM}$. We introduce a formalism of Operator Mechanics (OM) based on a noncommutative Poisson, symplectic and noncommutative differential structures. OM serves as a pre-quantum algebra from which algebraic structures relevant to real-world classical and quantum mechanics follow. In particular, $ψ_{KvN}$ and $ψ_{QM}$ are both consequences of this pre-quantum formalism. No a priori Hilbert space is needed. OM admits an algebraic notion of operator expectation values without invoking states. A phase space bundle ${\cal E}$ follows from this. $ψ_{KvN}$ and $ψ_{QM}$ are shown to be sections in ${\cal E}$. The difference between $ψ_{KvN}$ and $ψ_{QM}$ originates from a quantization map interpreted as "twisting" of sections over ${\cal E}$. We also show that the Schrödinger equation is obtained from the Koopman-von Neumann equation. What this suggests is that neither the Schrödinger equation nor the quantum wave function are fundamental structures. Rather, they both originate from a pre-quantum operator algebra. Finally, we comment on how entanglement between these operators suggests emergence of space; and possible extensions of this formalism to field theories.
We introduce new topological quantum invariants of compact oriented 3-manifolds with boundary where the boundary is a disjoint union of two identical surfaces. The invariants are constructed via surgery on manifolds of the form $F \times I$ where $I$ denotes the unit interval. Since virtual knots and links are represented as links in such thickened surfaces, we are able also to construct invariants in terms of virtual link diagrams (planar diagrams with virtual crossings). These invariants are the first meaningful, nontrivial, and calculable examples of quantum invariants of 3-manifolds with non-vacuous boundary. We give a new invariant of classical links in the 3-sphere in the following sense: Consider a link $L$ in $S^3$ of two components. The complement of a tubular neighborhood of $L$ is a manifold whose boundary consists in two copies of a torus. Our invariants apply to this case of bounded manifold and give new invariants of the given link of two components. Invariants of knots are also obtained.
This paper is dedicated to the memory of Zbigniew Oziewicz, to his generosity, intelligence and intensity in the search that is science and mathematics. The paper begins with a basic construction that produces Clifford algebras inductively, starting with a base algebra A that is associative and has an involution. This construction is an analog of the Cayley-Dickson Construction that produces the complex numbers, quaternions and octonions starting from the real numbers. Our basic construction always produces associative algebras and can be iterated an indefinite number of times. We generalize the basic construction to a group theoretic construction where a group G acts on the algebra A, and show how this group theoretic construction is related to matrix algebras. The paper then concentrates on applications of this algebra to the Dirac Equation and the Majorana-Dirac Equation.
We discuss Vassiliev invariants for virtual knots, expanding upon the theory of quantum virtual knot invariants developed in arXiv:1509.00578. In particular, following the theory of quantum invariants we work with 'rotational' virtual knots. We define chord diagrams, weight systems, and give examples of Lie algebra weight systems of rotational virtual knots. We end with a discussion of extended quantum invariants, which capture information that standard quantum invariants of rotational virtuals cannot.
We study invariants of virtual graphoids, which are virtual spatial graph diagrams with two distinguished degree-one vertices modulo graph Reidemeister moves applied away from the distinguished vertices. Generalizing previously known results, we give topological interpretations of graphoids. There are several applications to virtual graphoid theory. First, virtual graphoids are suitable objects for studying knotted graphs with open ends arising in proteins. Second, a virtual graphoid can be thought of as a way to represent a virtual spatial graph without using as many crossings, which can be advantageous for computing invariants.
We define a second Steenrod square for virtual links, which is stronger than Khovanov homology for virtual links, toward constructing Khovanov-Lipshitz-Sarkar stable homotopy type for virtual links. This induces the first meaningful nontrivial example of the second Steenrod square operator on the Khovanov homology for links in a 3-manifold other than the 3-sphere.
Knotted vortices such as those produced in water by Kleckner and Irvine tend to transform by reconnection to collections of unknotted and unlinked circles. The reconnection number $R(K)$ of an oriented knot of link $K$ is the least number of reconnections (oriented re-smoothings) needed to unknot/unlink $K$. Putting this problem into the context of knot cobordism, we show, using Rasmussen's Invariant that the reconnection number of a positive knot is equal to twice the genus of its Seifert spanning surface. In particular an $(a,b)$ torus knot has $R = (a-1)(b-1).$ For an arbitrary unsplittable positive knot or link $K$, $R(K) = c(K) - s(K) + 1$ where $c(K)$ is the number of crossings of $K$ and $s(K)$ is the number of Seifert circles of $K.$ Examples of vortex dynamics are illustrated in the paper.
This paper is an introduction to Khovanov homology, starting with the Kauffman bracket state summation, emphasizing the Bar-Natan Canopoloy and tangle cobordism approach. The paper discusses a simplicial approach to Khovanov homology and a quantum model for it so that the graded Euler characteristic that produces the Jones polynomial from Khovanov homology becomes the trace of a unitary transformation on a Hilbert space associated with the Khovanov Homology.
In the prequel of this paper, Kauffman and Ogasa introduced new topological quantum invariants of compact oriented 3-manifolds with boundary where the boundary is a disjoint union of two identical surfaces. The invariants are constructed via surgery on manifolds of the form $F \times I$ where $I$ denotes the unit interval. Since virtual knots and links are represented as links in such thickened surfaces, we are able also to construct invariants in terms of virtual link diagrams (planar diagrams with virtual crossings). These invariants are new, nontrivial, and calculable examples of quantum invariants of 3-manifolds with non-vacuous boundary. Since virtual knots and links are represented by embeddings of circles in thickened surfaces, we refer to embeddings of circles in the 3-sphere as {\it classical links}. Classical links are the same as virtual links that can be represented in a thickened 2-sphere and it is a fact that classical links, up to isotopy, embed in the collection of virtual links taken up to isotopy. We give a new invariant of classical links in the 3-sphere in the following sense: Consider a link $L$ in $S^3$ of two components. The complement of a tubular neighborhood of $L$ is a manifold whose boundary consists in two copies of a torus. Our invariants apply to this case of bounded manifold and give new invariants of the given link of two components. Invariants of knots are also obtained. In this paper we calculate the topological quantum invariants of some examples explicitly. We conclude from our examples that our invariant is new and strong enough to distinguish some classical knots from one another. We also explain how different Our topological quantum invariants of 3-manifolds with boundary and the Reshetikhin-Turaev invariants are. (See the body for detail).