arXiv research

Proves special alternating knots cannot be decomposed as non-trivial band sums.

Proves special alternating knots can't have cosmetic crossings.

New bounds on ropelength for special alternating knots.

Characterizes a specific type of alternating knot.

New invariant for special alternating links based on graph Laplacian.

Special knots with many twists have no certain type of surgery.

Study negative definite spin fillings of knot covers.

We solve a century-old conjecture about Alexander polynomials of special alternating links.

We give a topological characterisation of alternating knot exteriors based on the presence of special spanning surfaces. This shows that alternating is a topological property of the knot exterior and not just a property of diagrams, answering an old question of Fox. We also give a characterisation of alternating link e…

This paper computes Kakimizu complexes for all 11 crossing prime alternating knots.

Study sharpens unlinking number bounds for special alternating links.

The study examines quasi-alternating surgeries on knots and their properties.

In an earlier paper, we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y, which is closely related to the Heegaard Floer homology of Y. In this paper we investigate some properties of these knot homology groups for knots in the three-sphere. We give a combinatorial description fo…

New cobordism maps for instanton knot homology help compute surgeries on knots.

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…

Bounds on knot polynomials for Lie superalgebras of type I.

We establish a characterization of alternating links in terms of definite spanning surfaces. We apply it to obtain a new proof of Tait's conjecture that reduced alternating diagrams of the same link have the same crossing number and writhe. We also deduce a result of Banks and Hirasawa-Sakuma about Seifert surfaces for…

It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman's Tutte polynomial of signed graphs for which describi…

Study SO(3)-knot states for torus complements, linking to simplicial volume.

In this thesis we work with Khovanov homology of links and its generalizations, as well as with the homology of graphs. Khovanov homology of links consists of graded chain complexes which are link invariants, up to chain homotopy, with graded Euler characteristic equal to the Jones polynomial of the link. Hence, it can…

The paper explores knots with equal bridge and braid index, conjecturing they have a unique equilibrium state.

Study on Fox's trapezoidal conjecture for specific alternating links.

Positive knots are minimal in a specific knot ordering.

Study shows bounds on volumes of weakly generalised alternating knots.

Study concordance of alternating torus knots to L-space knots.

The study calculates and analyzes alternating surgeries for various knots.

A conjecture proposed by J. Tripp in 2002 states that the crossing number of any knot coincides with the canonical genus of its Whitehead double. In the meantime, it has been established that this conjecture is true for a large class of alternating knots including $(2, n)$ torus knots, $2$-bridge knots, algebraic alter…

We extend Howie's characterization of alternating knots to give a topological characterization of toroidally alternating knots, which were defined by Adams. We provide necessary and sufficient conditions for a knot to be toroidally alternating. We also give a topological characterization of almost-alternating knots whi…

Algorithm finds ribbon disks for alternating knots, resolving sliceness for most prime knots.

Recent advances in Quantum Topology assign $q$-series to knots in at least three different ways. The $q$-series are given by generalized Nahm sums (i.e., special $q$-hypergeometric sums) and have unknown modular and asymptotic properties. We give an efficient method to compute those $q$-series that come from planar gra…

Bankwitz characterized an alternating diagram representing the trivial knot. A non-alternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize an almost alternaing diagram representing the trivial knot. As a corollary we determine an unknotting number one alter…

New invariant for prime alternating knots from error-correcting codes

The study proves large alternating Montesinos knots cannot have purely cosmetic surgeries.

This paper finds all prime alternating knots with minimal warping degree two.

It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is 1. We extend this result to almost positive links and partly identify the 3 following coefficients for special types of positive links. We also give counterexamples to the Jones polynomia…

We prove that if an alternating knot has unknotting number one, then there exists an unknotting crossing in any alternating diagram. This is done by showing that the obstruction to unknotting number one developed by Greene in his work on alternating 3-braid knots is sufficient to identify all unknotting number one alte…

Generalizing Howie and Greene's characterization of alternating knots, we give a topological characterization of almost alternating knots.

In this paper we define Crossing Change Alternating Knots (CCA knots) and their generalization: $k$-CCA knots.

Automatic computation speeds up crosscap number calculation for alternating knots.

Study bounds on cusp volumes of alternating knots on surfaces.

Jones slopes detect figure eight knot, and characterize alternating knots.

Computing polynomial invariants for knots and links using braid representations relies heavily on finding the trace of Hecke algebra elements. There is no easy method known for computing the trace and hence it becomes difficult to compute the known polynomial invariants of knots using their braid representations. In th…

We create Lefschetz fibrations for knot traces of specific types of knots.

The aim of this article is to detect new classes of quasi-alternating links. Quasi-alternating links are a natural generalization of alternating links. Their knot Floer and Khovanov homology are particularly easy to compute. Since knot Floer homology detects the genus of a knot as well as whether a knot is fibered, as …

This is the first in a series of four papers wherein we enumerate all prime alternating knots and links. In this first paper, we introduce four operators on knots and show that, when used according to very simple rules on the prime alternating knots of n crossings, the set of all prime alternating knots of n+1 crossing…

Cyclic covers of knots uniquely determine the original knot.

The article confirms two quasi-alternating surgeries for 9 asymmetric L-space knots.

It was previously shown by the second author that every knot in $S^3$ is ambient isotopic to one component of a two-component, alternating, hyperbolic link. In this paper, we define the alternating volume of a knot $K$ to be the minimum volume of any link $L$ in a natural class of alternating, hyperbolic links such tha…