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Louis H. Kauffman

Publications and source records attributed to Louis H. Kauffman.

At least 19 recordsLinked to original sources
arXiv2026

A counterexample for the polar conjecture of Spencer-Brown

In 1976, George Spencer-Brown announced a proof of the four color theorem, using operations on Tait colorings for trivalent plane graphs. In subsequent work he formulated these operations in terms of an algorithm that he called a parity-pass and claimed that when the parity pass algorithm is performed on a non-polar pentagon region, it necessarily terminates in an edge coloring that is extendable to the entire graph. We provide here a counterexample to show that this claim is false. We then raise questions related to the existence of this sort of counterexample.

math.CO↗
arXiv2026

Probabilistic pseudo knot theory

We develop the theory of \emph{probabilistic pseudo knots}, providing a framework for modeling knot diagrams with unresolved crossing information. Pseudo knot diagrams generalize classical diagrams by allowing certain crossings to remain unspecified; in the probabilistic setting, each such \emph{pre-crossing}, namely a crossing with undetermined over--under information, is assigned a probability describing the likelihood of resolving as a positive crossing, with complementary probability assigned to the negative resolution. This induces a probability distribution on complete classical resolutions and, by aggregation, a distribution on classical knot types, capturing uncertainty arising in physical, biological, and computational contexts. We introduce \emph{probabilistic equivalence}, defined via total variation distance between resolution distributions, and extend classical numerical quantities such as writhe and linking number to this setting. We also develop new probabilistic constructions, including the probabilistic chirality index, minimal resolution genus, probabilistic Seifert surface distributions, and polynomial invariants extending the Kauffman bracket. We further discuss matrix-based constructions, including probabilistic Seifert and Goeritz-type matrices, as well as probabilistic surgery producing distributions over 3-manifolds. Finally, we discuss potential applications in molecular biology, materials science, and computational topology.

math.GT↗
arXiv2026

The Q-Calculus: A Quaternion-Based Laws of Form System

This paper introduces a Laws of Form version of the Quaternions. We call this the Q-Calculus, a 16-valued extension of Laws of Form (LoF) which is closely related to the BF Calculus (where we have a single square root of the mark) and the concept of the square root of negation (related to the the square root of minus one). We construct Q as a system of LoF mark operators acting on 4-tuples, and prove that the set of eight operators in Q is isomorphic to the quaternion group, which is non-commutative. We give a novel proof of several of Q's distribution laws using non-commutative logic gates. We indicate how to represent Q as braids by associating elementary braids to square roots of negation. This results in a very concise representation of Q as LoF braids. We end the paper with an indication of how we can represent the Artin braid group in LoF and how we can generalize our work with the quaternions to Clifford algebras.

math.LO↗
arXiv2026

The Mereon System, the 600-Cell, and the Exceptional Algebras $E_6$, $E_7$, $E_8$: Exact Correspondence via $H_3 \subset H_4$ Symmetry and the Eigenform Loop

This work concerns how the three-dimensional polyhedral Mereon structure (the 120 polyhedron) is the precise projection from four-space of the 600-cell, an analogue in four-dimensional space of a regular solid. The 600-cell is made from 120 copies of a dodecahedron that are fitted together so that each dodecahedral face is matched to the face of another dodecahedron (much as the pentagonal faces of the dodecahedron are matched along their edges). Thus this essential part of the Mereon structure is a projection from a higher-dimensional space of an even more symmetrical entity. The theme that three-dimensional structures, earthly structures, networked structures, structures involved in our understanding and communication, would be or should be seen as projections from a higher-dimensional whole is part of perennial philosophy. Here we are seeing an instantiation of this theme and the dreams with which it is allied. The 600-cell and its associated geometries have been studied for some time by mathematicians and by physicists for relations with geometry, topology, knot theory, particle physics and even cosmology and string theory. It is more than exciting that there is a direct connection of the Mereon System with the 600-cell and the wide-ranging conversation with which it is associated. We expect much more from this connection as the search goes on. The present paper makes this connection precise. We establish the exact correspondence between the Mereon System and the 600-cell, and show how the nested architecture realises all three exceptional Lie algebras $E_6$, $E_7$, $E_8$.

math.GR↗
arXiv2026

Wormhole Nucleation via Topological Surgery in Lorentzian Geometry

We construct a model for the nucleation of a wormhole within a Lorentzian spacetime by employing techniques from topological surgery and Morse theory. In our framework, a 0-surgery process describes the neighborhood of the nucleation point inside a compact region of spacetime, yielding a singular Lorentzian cobordism that connects two spacelike regions with different topologies. To avoid the singularity at the critical point of the Morse function, we employ the Misner trick of taking a connected sum with a closed 4-manifold -- namely $\mathbb{CP}^{2}$ -- to obtain an everywhere nondegenerate Lorentzian metric. This connected sum replaces the naked singularity with a region containing closed timelike curves. The obtained spacetime is nonsingular, but violates all the standard energy conditions. Our construction, thus, shows that a wormhole can be "created" without singularities in classical general relativity.

gr-qc↗
arXiv2026

The Penrose-Kauffman Polynomial

For any cubic graph in a closed orientable surface and a perfect matching, the Penrose-Kauffman polynomial is a sum of chromatic polynomials of a collection of associated graphs. A knot-theoretic perspective affords elementary proofs of old and new results about the polynomial. The Four Color Theorem is shown to be equivalent to a statement about 3-coloring alternating link diagrams in the plane that are reduced and have no bigon regions.

math.GT↗
arXiv2026

Exotic 4-manifolds and Khovanov-Lipshitz-Sarkar homotopy type

We introduce a new diffeomorphism invariant of smooth compact oriented 4-manifolds $X$ with a framed oriented 1-link $L$ in the boundary, where $L$ may be the empty set, and call it {\it Khovanov-Lipshitz-Sarkar skein lasagna homotopy type} or {\it KLS lasagna homotopy type} $\mathcal E^{LS}_0(X,L)$. Our invariant assigns to a smooth structure a stable homotopy type of a CW complex. Our new invariant is not weaker than KR lasagna module, which were defined by Morrison, Walker and Wedrich. For a pair $(X,L)$ such that $L\neq\emptyset$, our new invariant, KLS lasagna homotopy type, is stronger than the Khovanov-Rozansky $\mathfrak{gl}_2$ skein lasagna modules or KR lasagna modules.

math.GT↗
arXiv2025

Topology and Algebra of Bonded Knots and Braids

In this paper we present a detailed study of \emph{bonded knots} and their related structures, integrating recent developments into a single framework. Bonded knots are classical knots endowed with embedded bonding arcs modeling physical or chemical bonds. We consider bonded knots in three categories (long, standard, and tight) according to the type of bonds, and in two categories, topological vertex and rigid vertex, according to the allowed isotopy moves, and we define invariants for each category. We then develop the theory of \emph{bonded braids}, the algebraic counterpart of bonded knots. We define the {\it bonded braid monoid}, with its generators and relations, and formulate the analogues of the Alexander and Markov theorems for bonded braids, including an $L$-equivalence for bonded braids. Next, we introduce \emph{enhanced bonded knots and braids}, incorporating two types of bonds (attracting and repelling) corresponding to different interactions. We define the enhanced bonded braid group and show how the bonded braid monoid embeds into this group. Finally, we study \emph{bonded knotoids}, which are open knot diagrams with bonds, and their closure operations, and we define the \emph{bonded closure}. We introduce \emph{bonded braidoids} as the algebraic counterpart of bonded knotoids. These models capture the topology of open chains with inter and intra-chain bonds and suggest new invariants for classifying biological macromolecules.

math.GT↗
arXiv2025

A robot that unknots knots

Consider a robot that remembers only the starting position and walks along a knot once on a knot diagram, switching every undercrossing it meets until it returns to the starting position. We observe that the robot produces an ascending diagram, and we provide a new combinatorial proof that every ascending or descending knot diagram can be transformed into the zero-crossing unknot diagram. Using the machinery developed from the combinatorial proof, we show that the minimal number of Reidemeister moves required for such a transformation is bounded above by (7C+1)C if the diagram has C crossings. Moreover, we provide a new alternative proof that there exist sequences of Reidemeister moves that do not increase the number of crossings and transform ascending or descending knot diagrams into zero-crossing unknot diagrams.

math.GT↗
arXiv2025

Fundamentals of cubic skein modules

Over the past thirty-seven years, the study of linear and quadratic skein modules has produced a rich and far-reaching skein theory, intricately connected to diverse areas of mathematics and physics, including algebraic geometry, hyperbolic geometry, topological quantum field theories, and statistical mechanics. However, despite these advances, skein modules of higher degree-those depending on more parameters than the linear and quadratic cases-have received comparatively little attention, with only a few isolated explorations appearing in the literature. In this article, we undertake a systematic study of the cubic skein module, the first representative of this broader class. We begin by investigating its structure and properties in the $3$-sphere, and then extend the analysis to arbitrary $3$-manifolds. The results presented here aim to establish a foundational framework for the study of higher skein modules, thereby extending the scope of skein theory beyond its classical domains. Furthermore, studying the structure of cubic skein modules may lead to new polynomial invariants of knots.

math.GT↗
arXiv2025

Knots in $\mathbb{R}P^3$

This paper studies knots in three dimensional projective space. Our technique is to associate a virtual link to a link in projective space so that equivalent projective links go to equivalent virtual links (modulo a special flype move). We apply techniques in virtual knot theory to obtain a Jones polynomial for projective links. We show that this is equivalent to the known Jones polynomial defined by Drobotukhina for them. We apply virtual Khovanov homology and the virtual Rasmussen invariant of Dye, Kaestner, and Kauffman to projective links. We compare this cohomology theory with the Khovanov type theory developed by Manolescu and Willis for projective knots. We show that these theories are essentially equivalent.

math.GT↗
arXiv2025

Primes Between Squares -- Commentary on Appendix 8 of Laws Of Form

This paper provides a commentary and guide to Appendix 8 of Laws Of Form, which is a chapter (appendix) on number theory in the book Laws of Form by Spencer-Brown. (Spencer-Brown,Laws Of Form,Revised Seventh English edition. Bohmeier Verlag. 2020) This chapter in the book provides Spencer-Brown's proofs of the conjecture that there are at least two prime numbers between any consecutive squared numbers. That there are primes between squares has been a conjecture in number theory since Legendre. In Spencer-Brown's appendix he gives his proofs of the conjecture. Those proofs are a highly original mixture of standard rigorous arguments and also some stated facts about the way numbers behave that would be considered conjectures by most number theorists. These phenomena are very interesting and constitute a deep observation about the nature of number itself. We intend that our guide will enable the reader to gain insight into Spencer-Brown's point of view and that our discussions will be of interest to anyone with curiosity about the theory of numbers.

math.NT↗
arXiv2025

The clock theorem for knotoids and linkoids

In this paper, we generalize the \textit{Clock Theorem} of Formal Knot Theory to knotoids in $S^2$. The clock theorem implies that clock states of a knotoid diagram form a lattice under transpositions. These states form the basis of many invariants of knotoids and linkoids including the Alexander polynomial, Mock Alexander polynomial and the Jones polynomial.

math.GT↗
arXiv2025

Fusion and Fission of Particle-like Chiral Nematic Vortex Knots

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.

cond-mat.soft↗
arXiv2025

Algebraic invariants of multi-virtual links

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.

math.GT↗
arXiv2025

Preons, Braid Topology, and Representations of Fundamental Particles

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.

physics.gen-ph↗