arXiv research

Study Murasugi sum in 4D for knotted surfaces, defining arborescent surfaces.

We use the methods of Hedden, Juhasz, and Sarkar to exhibit a set of arborescent knots that bound large numbers of non-isotopic minimal genus spanning surfaces. In particular, we describe a sequence of prime knots K_{n} which will bound at least 2^{2n-1} non-isotopic minimal spanning surfaces of genus n.

Paper connects link determinants to polytopes and arborescence counts.

In this paper, we study the hyperbolicity of arborescent tangles and arborescent links. We will explicitly determine all essential surfaces in arborescent tangle complements with non-negative Euler characteristic, and show that given an arborescent tangle T, the complement X(T) is non-hyperbolic if and only if T is a r…

Positive Thompson links are arborescent tangles.

Locally finite complexes with polyhedral $CAT(0)$ metrics are arborescent.

The study proves a conjecture about arborescent links with many twigs.

Unified framework for gradient estimation in combinatorial spaces.

Method determines trace-free SL(2,C) reps for arborescent links.

Study character varieties of arborescent knots and hyperbolic knots.

Formula for arborescent link tails using theta functions.

A Dehn surgery on a knot $K$ in $S^3$ is exceptional if it produces a reducible, toroidal or Seifert fibred manifold. It is known that a large arborescent knot admits no such surgery unless it is a type II arborescent knot. The main theorem of this paper shows that up to isotopy there are exactly three large arborescen…

Paper computes Alexander polynomials for arborescent links.

New algebra counts components of arborescent knots and links.

Authors show that genus defects of Hopf arborescent links are decidable.

This study analyzes information flow networks in Chinese stock sectors using transfer entropy.

We realize a given (monic) Alexander polynomial by a (fibered) hyperbolic arborescent knot and link of any number of components, and by infinitely many such links of at least 4 components. As a consequence, a Mahler measure minimizing polynomial, if it exists, is realized as the Alexander polynomial of a fibered hyperb…

Study $d$-invariants of L-space double branched covers of arborescent links.

For a positive Hopf plumbed arborescent Seifert surface $S$, we study the set of Hopf bands $H\subset S$, up to homology and up to the action of the monodromy. The classification of Seifert surfaces for which this set is finite is closely related to the classification of finite Coxeter groups.

Four constructions of Seifert surfaces - Hopf plumbing, arborescent plumbing, basketry, and T-bandword handle decomposition - are described, and some interrelationships found, e.g.: arborescent Seifert surfaces are baskets; Hopf-plumbed baskets are precisely homogeneous T-bandword surfaces. A Seifert surface is Hopf-pl…

We briefly review 3-dimensional untwisted Dijkgraaf-Witten theory over a finite group $Γ$, and present a method of computing untwisted Dijkgraaf-Witten invariants for arborescent links. Some explicit formulas are given when $Γ=\mathbb{Z}/p\mathbb{Z}\rtimes\mathbb{Z}/(p-1)\mathbb{Z}$ for an odd prime $p$.

Algorithm reconstructs conserved networks from flow data.

Constructs a new graded variety from algebraic data.

This paper describes a way to subdivide a 3-manifold into angled blocks, namely polyhedral pieces that need not be simply connected. When the individual blocks carry dihedral angles that fit together in a consistent fashion, we prove that a manifold constructed from these blocks must be hyperbolic. The main application…

New links identified with Alexander polynomial signs to detect satellite links.

Arborescent knots are the ones which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is enough for lifting topological description to the level of effective analytical formulas. The paper describes the origin and struc…

This article is devoted to the study of prime alternating +achiral knots. In the case of arborescent knots, we prove in +AAA Visibility Theorem 5.1, that the symmetry is visible on a certain projection (not necessarily minimal) and that it is realised by a homeomorphism of order 4. In the general case (arborescent or n…

In this paper, we give an isotopy classification of 3-bridge spheres of 3-bridge arborescent links, which are not Montesinos links. To this end, we prove a certain refinement of a theorem of J.S. Birman and H.M. Hilden on the relation between bridge presentations of links and Heegaard splittings of 3-manifolds. In the …

KNTZ trick simplifies knot polynomial calculations for twist knots.

Researchers extend knot theory formulas to non-rectangular cases.

This article is solicited by C.\ Adams for a special issue of {\it Chaos, Solitons and Fractals\/} devoted to knot theory and its applications. We present some recent results about Dehn surgeries on arborescent knots and links.

New method extracts Racah matrices from antiparallel double-braid knots.

Roberts proved that a family of alternating, arborescent, prime knots each have at least $2^{2n-1}$ distinct minimal genus Seifert surfaces, where $n$ is the genus of the knot in question. We give a subfamily of these knots that have exactly this many minimal genus Seifert surfaces.

We give a complete classification of exceptional surgeries on hyperbolic alternating knots in the 3-sphere. As an appendix, we also show that the Montesinos knots M (-1/2, 2/5, 1/(2q + 1)) with q at least 5 have no non-trivial exceptional surgeries. This gives the final step in a complete classification of exceptional …

By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in p…

Let Πbe a link projection in S^2. John Conway and later Francis Bonahon and Larry Siebenmann undertook to split $Π$ into canonical pieces. These pieces received different names: basic or polyhedral diagrams on one hand, rational, algebraic, bretzel, arborescent diagrams on the other hand. This paper proposes a thorough…

Exceptional Dehn surgeries on arborescent knots have been classified except for Seifert fibered surgeries on Montesinos knots of length 3. There are infinitely many of them as it is known that 4n+6 and 4n+7 surgeries on a (-2, 3, 2n+1) pretzel knot are Seifert fibered. It will be shown that there are only finitely many…

Study proves meridional rank conjecture for certain complex links.

We give a counterexample to the Kawauchi conjecture on the Conway polynomial of achiral knots which asserts that the Conway polynomial $C(z)$ of an achiral knot satisfies the splitting property $C(z)=F(z)F(-z)$ for a polynomial $F(z)$ with integer coefficients. We show that the Bonahon-Siebenmann decomposition of an ac…

Researchers derive exclusive Racah matrix for complex representation with non-trivial multiplicities.

We study when the Thurston norm is detected by twisted Alexander polynomials associated to representations of the 3-manifold group to SL(2, C). Specifically, we show that the hyperbolic torsion polynomial determines the genus for a large class of hyperbolic knots in the 3-sphere which includes all special arborescent k…

This work extends SVM error bounds to weighted SVM and introduces hyperparameter selection methods.

Authors determine the SL(2,C) character variety of a specific knot without computer aid.

New spanning 3-disks found for unlink in 4-sphere.

The cosmetic crossing conjecture (also known as the "nugatory crossing conjecture") asserts that the only crossing changes that preserve the oriented isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery characterization of the unknot to prove this conjecture for knots in integer homology sphere…

Develops tests for Markowitz stochastic dominance spanning using saddle points.

Sharp bounds for spanning tree entropy in planar lattices.

Totally geodesic surfaces found in knots and links.