arXiv research

A new quandle from link modules helps identify link properties.

Enhanced Alexander module detects linking numbers in links.

We characterize the first Alexander Z[Z]-modules of ribbon surface-links in the 4-sphere fixing the number of components and the total genus, and then the first Alexander Z[Z]-modules of surface-links in the 4-sphere fixing the number of components. Using the result of ribbon torus-links, we also characterize the first…

Researchers compute link surgery modules for 2-component L-space links.

Enhanced bikei modules distinguish unoriented and non-orientable surface-links.

Enhances knot and link invariants using quandle modules.

Introduces Floer lasagna modules using link Floer homology.

Study disproves finiteness conjecture for a specific link's skein module.

Classifies modules of surface-knots in terms of their properties.

We construct examples of knots that have isomorphic nth-order Alexander modules, but non-isomorphic nth-order linking forms, showing that the linking forms provide more information than the modules alone. This generalizes work of Trotter, who found examples of knots that have isomorphic classical Alexander modules, but…

For a ring $R$, we denote by $R[\mathcal L]$ the free $R$-module spanned by the isotopy classes of singular links in $\mathbb S^3$. Given two invertible elements $x,t \in R$, the HOMFLY-PT skein module of singular links in $\mathbb S^3$ (relative to the triple $(R,t,x)$) is the quotient of $R[\mathcal L]$ by local rela…

We study the concept of the fourth skein module of 3-manifolds, that is a skein module based on the skein relation b_0L_0 + b_1L_1 + b_2L_2 + b_3L_3 = 0 and a framing relation L^{(1)} = aL, where a, b_0, b_3 are invertible. We introdule the concept of n-algebraic tangles (and links) and analyze the skein module for 3-a…

Link Floer homology detects split links.

New invariants for surface-links using biquandle modules.

Study shows Conway mutation preserves a specific link invariant.

We explain how Queffelec-Sartori's construction of the HOMFLY-PT link polynomial can be interpreted in terms of parabolic Verma modules for $\mathfrak{gl}_{2n}$. Lifting the construction to the world of categorification, we use parabolic 2-Verma modules to give a higher representation theory construction of Khovanov-Ro…

We compute the Khovanov lasagna module of S²×S², confirming a conjecture.

Link homology theories connect to 4-manifold invariants and TQFTs.

Constructs the medial quandle from link's peripheral structure.

Modified proof for a broader class of links.

Minor typographical errors fixed. Cochran constructed many links with Alexander module that of the unlink and some nonvanishing Milnor invariants, using as input commutators in a free group and as an invariant the longitudes of the links. We present a different and conjecturally complete construction, that uses element…

We introduce higher skein modules of links generalizing the Conway skein module. We show that these modules are closely connected to the HOMFLY polynomial.

Study uses graph Laplacians to analyze surface links.

New links found that aren't concordant to any with a single unknotted component.

Research shows Khovanov homology can identify split links.

Defines super projective modules and explores their properties.

Innovative rack theory applied to Legendrian links.

Walter Neumann showed that the topology of a ``regular'' algebraic curve V in C^2 is determined up to proper isotopy by some link in S^3 called the link at infinity of V. In this note, we compute the Alexander module over C[t^{\pm 1}] of any such link at infinity.

Worldsheet skein D-module for Hopf link conormal uniquely determines partition functions.

We define an invariant of tangles and framed tangles given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks, quandles, rack and quandle cocycles, 2-crossed modules and braided crossed modules. We…

We calculate the Kauffman bracket skein module (KBSM) of the complement of all two-bridge links. For a two-bridge link, we show that the KBSM of its complement is free over the ring $\BC[t^{\pm 1}]$ and when reducing $t=-1$, it is isomorphic to the ring of regular functions on the character variety of the link group.

New modules derived from Khovanov homology for links.

We introduce an associative algebra Z[X,S] associated to a birack shadow and define enhancements of the birack counting invariant for classical knots and links via representations of Z[X,S] known as shadow modules. We provide examples which demonstrate that the shadow module enhanced invariants are not determined by th…

New 4-manifold invariants from Khovanov-Rozansky link homology.

We show that the Kauffman bracket skein modules of certain manifolds obtained from integral surgery on a (2,2b) torus link are finitely generated, and list the generators for select examples.

Frobenius extensions play a central role in the link homology theories based upon the sl(n) link variants, and each of these Frobenius extensions may be recast geometrically via a category of marked cobordisms in the manner of Bar-Natan. Here we explore a large family of such marked cobordism categories that are releva…

In this paper the properties of the Kauffman bracket skein module of $L(p,q)$ are investigated. Links in lens spaces are represented both through band and disk diagrams. The possibility to transform between the diagrams enables us to compute the Kauffman bracket skein module on an interesting class of examples consisti…

New polynomial invariants for knots and links.

This paper generalize [7](math.GT/0601291): We construct new links invariants from g, a type I basic classical Lie superalgebra. The construction uses the existence of an unexpected replacement of the vanishing quantum dimension of typical module. Using this, we get a multivariable link invariant associated to any one …

We compute two-term skein modules of framed oriented links in oriented 3-manifolds. They contain the self-writhe and total linking number invariants of framed oriented links in a universal way. The relations in a natural presentation of the skein module are interpreted as monodromies in the space of immersions from cir…

Formulae for 1-3 handle attachments in 4-manifolds.

Niebrzydowski tribrackets are ternary operations on sets satisfying conditions obtained from the oriented Reidemeister moves such that the set of tribracket colorings of an oriented knot or link diagram is an invariant of oriented knots and links. We introduce tribracket modules analogous to quandle/biquandle/rack modu…

Algebra Situs is a branch of mathematics which has its roots in Jones' construction of his polynomial invariant of links and Drinfeld's work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum g…

In arXiv:math/0508510, Rasmussen observed that the Khovanov-Rozansky homology of a link is a finitely generated module over the polynomial ring generated by the components of this link. In the current paper, we study the module structure of the middle HOMFLYPT homology, especially the Betti numbers of this module. For …

The paper extends Alexander invariant and medial quandle equivalence to links.

A new method to derive presentations of skein modules is developed. For the case of homotopy skein modules it will be shown how the topology of a 3-manifold is reflected in the structure of the module. The freeness problem for q-homotopy skein modules is solved, and a natural skein module related to linking numbers is …

Researchers computed Kauffman bracket skein modules of specific Seifert manifolds.

Following the recent work by Chan, and by Morton and Hadji on the Homflypt polynomials of some generalized Hopf links, we investigate the Kauffman polynomials of generalized Hopf links. By studying the Kauffman skein module of the solid torus S^1\times D^2, we establish a similar skein map on the Kauffman skein module …