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94 papers · 1 filter
Topology and Algebra of Bonded Knots and Braids
Ioannis Diamantis, Louis H. Kauffman, Sofia Lambropoulou
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 k…
Drilling hyperbolic groups
Daniel Groves, Peter Haïssinsky, Jason F. Manning +3
Given a hyperbolic group and a maximal infinite cyclic subgroup , we define a {\it drilling of along }, which is a relatively hyperbolic group pair $(…
Realizing pairs of multicurves as cylinders on translation surfaces
Juliet Aygun, Janet Barkdoll, Aaron Calderon +2
Any pair of intersecting cylinders on a translation surface is "coherent," in that the geometric and algebraic intersection numbers of their core curves are equal (up to sign). In…
Volume and topology of bounded and closed hyperbolic 3-manifolds, II
Jason DeBlois, Peter B. Shalen
Let be a compact, orientable hyperbolic 3-manifold whose boundary is a connected totally geodesic surface of genus . If has Heegaard genus at least , then its volume…
The Virtual Spectrum of Linkoids and Open Curves in 3-space
Kasturi Barkataki, Louis H. Kauffman, Eleni Panagiotou
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 recent…
Centrality and the KRH Invariant
Louis H Kauffman, David Radford, Stephen Sawin
The purpose of this paper is to discuss the categorical structure for a method of defining quantum invariants of knots, links and three-manifolds. These invariants can be defined i…
Vassiliev measures of complexity for open and closed curves in 3-space
Eleni Panagiotou, Louis H. Kauffman
In this manuscript we define Vassiliev measures of complexity for open curves in 3-space. These are related to the coefficients of the enhanced Jones polynomial of open curves in 3…
Virtual Multicrossings and Petal Diagrams for Virtual Knots and Links
Colin Adams, Chaim Even-Zohar, Jonah Greenberg +6
Multicrossings, which have previously been defined for classical knots and links, are extended to virtual knots and links. In particular, petal diagrams are shown to exist for all…
Quantum invariants of knotoids
Neslihan Gugumcu, Louis H. Kauffman
In this paper, we construct quantum invariants for knotoid diagrams in . The diagrams are arranged with respect to a given direction in the plane ({\it Morse knotoids…
Embedding closed hyperbolic 3-manifolds in small volume hyperbolic 4-manifolds
Michelle Chu, Alan W. Reid
In this paper we study existence and lack thereof of closed embedded orientable co-dimension one totally geodesic submanifolds of minimal volume cusped orientable hyperbolic manifo…
Knot polynomials of open and closed curves
Eleni Panagiotou, Louis H. Kauffman
In this manuscript we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket pol…
A hyperbolic counterpart to Rokhlin's cobordism theorem
Michelle Chu, Alexander Kolpakov
The purpose of the present paper is to prove existence of super-exponentially many compact orientable hyperbolic arithmetic -manifolds that are geometric boundaries of compact o…
The generalized Yamada polynomials of virtual spatial graphs
Qingying Deng, Xian'an Jin, Louis H. Kauffman
Classical knot theory can be generalized to virtual knot theory and spatial graph theory. In 2007, Fleming and Mellor combined virtual knot theory and spatial graph theory to form,…
Graphical virtual links and a polynomial of signed cyclic graphs
Qingying Deng, Xian'an Jin, Louis H Kauffman
For a signed cyclic graph G, we can construct a unique virtual link L by taking the medial construction and convert 4-valent vertices of the medial graph to crossings according to…
Majorana Fermions and Representations of the Braid Group
Louis H Kauffman
In this paper we study unitary braid group representations associated with Majorana Fermions. Majorana Fermions are represented by Majorana operators, elements of a Clifford algebr…
Volume and Homology for Hyperbolic -Orbifolds, I
Peter B. Shalen
Let be a closed, orientable, hyperbolic 3-orbifold whose singular set is a link, and such that contains no hyperbolic triangle group. We show t…
Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations
David Dumas, Andrew Sanders
We study the topology and geometry of compact complex manifolds associated to Anosov representations of surface groups and other hyperbolic groups in a complex semisimple Lie group…
Spectral gap of scl in free products
Lvzhou Chen
Let be a free product of torsion-free groups, and let be any element not conjugate into a . Then scl. This generalizes, and gives a new pro…
Coarse and fine geometry of the Thurston metric
David Dumas, Anna Lenzhen, Kasra Rafi +1
We study the geometry of the Thurston metric on the Teichmüller space of hyperbolic structures on a surface . Some of our results on the coarse geometry of this…
On the complexity of finite subgraphs of the curve graph
Edgar A. Bering, Gabriel Conant, Jonah Gaster
We say a graph has property when it is an induced subgraph of the curve graph of a surface of genus with punctures. Two well-known graph invariants, the…
Applications of fast triangulation simplification
Mark C. Bell, Richard C. H. Webb
We describe a new algorithm to compute the geometric intersection number between two curves, given as edge vectors on an ideal triangulation. Most importantly, this algorithm runs…
Tangle Insertion Invariants for Pseudoknots, Singular Knots, and Rigid Vertex Spatial Graphs
Allison Henrich, Louis H. Kauffman
The notion of a pseudoknot is defined as an equivalence class of knot diagrams that may be missing some crossing information. We provide here a topological invariant schema for pse…
Orderability and Dehn filling
Marc Culler, Nathan M. Dunfield
Motivated by conjectures relating group orderability, Floer homology, and taut foliations, we discuss a systematic and broadly applicable technique for constructing left-orders on…
Chern-Simons Theory, Vassiliev Invariants, Loop Quantum Gravity and Functional Integration Without Integration
Louis H. Kauffman
This paper is an exposition of the relationship between Witten's Chern-Simons functional integral and the theory of Vassiliev Invariants of knots and links in three dimensional spa…