相关论文: Inequivalent surface-knots with the same knot quan…
The nonorientable 4-genus is an invariant of knots which has been studied by many authors, including Gilmer and Livingston, Batson, and Ozsv\'{a}th, Stipsicz, and Szab\'{o}. Given a nonorientable surface $F \subset B^4$ with $\partial F =…
A classification of spanning surfaces for alternating links is provided up to genus, orientability, and a new invariant that we call aggregate slope. That is, given an alternating link, we determine all possible combinations of genus,…
Quandle cocycles are constructed from extensions of quandles. The theory is parallel to that of group cohomology and group extensions. An interpretation of quandle cocycle invariants as obstructions to extending knot colorings is given, and…
A polynomial is presented that models a topological knot in a unique manner. It distinguishes all types of knots including the orientation and has a group theory interpretation. The topologies may be labeled via a number, which upon a base…
The classical abelian invariants of a knot are the Alexander module, which is the first homology group of the the unique infinite cyclic covering space of S^3-K, considered as a module over the (commutative) Laurent polynomial ring, and the…
The knot quandle is a complete invariant for oriented classical knots in the $3$-sphere up to orientation. Eisermann computed the second quandle homology group of the knot quandle and showed that it characterizes the unknot. In this paper,…
We prove that the Turaev--Viro invariants of the two surface bundles over the circle coincide for every spherical fusion category if the surface group is procongruently conjugacy separable and there exists a regular profinite isomorphism…
Expanding on work by Conway, Orson, and Powell, we study the isotopy classes rel. boundary of nonorientable, compact, locally flatly embedded surfaces in $D^4$ with knot group $\mathbb{Z}_2$. In particular we show that if two such surfaces…
We establish inequalities that constrain the genera of smooth cobordisms between knots in 4-dimensional cobordisms. These "relative adjunction inequalities" improve the adjunction inequalities for closed surfaces which have been…
In this paper, a theory of quandle rings is proposed for quandles analogous to the classical theory of group rings for groups, and interconnections between quandles and associated quandle rings are explored.
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…
We describe a procedure for creating infinite families of hyperbolic knots having unique minimal genus Seifert surface. A large subset of these knots have the further property that the surface cannot be the sole compact leaf of a depth one…
We introduce a special class of knots, called global knots, in F^2 x R and we construct new isotopy invariants, called T-invariants, for global knots. Some T-invariants are of finite type but they cannot be extracted from the generalized…
J. Boyle classified 1-handles attached to surface-knots, that are closed and connected surfaces embedded in the Euclidean 4-space, in the case that the surfaces are oriented and 1-handles are orientable with respect to the orientations of…
We present a new, practical algorithm to test whether a knot complement contains a closed essential surface. This property has important theoretical and algorithmic consequences; however, systematically testing it has until now been…
This article is concerned with locally flatly immersed surfaces in simply-connected $4$-manifolds where the complement of the surface has fundamental group $\mathbb{Z}$. Once the genus and number of double points are fixed, we classify such…
The fundamental quandle is a powerful invariant of knots and links, but it is difficult to describe in detail. It is often useful to look at quotients of the quandle, especially finite quotients. One natural quotient introduced by Joyce is…
A number of results for the level-rank duality of $G(N)_K$ $\leftrightarrow$ $G(K)_N$ Chern-Simons theory are summarized, with emphasis on the applications to knot and link invariants. Explicit examples for $SU(2)_K$ $\leftrightarrow$…
In this paper we define and investigate Z/2-homology cobordism invariants of Z/2-homology 3-spheres which turn out to be related to classical invariants of knots. As an application we show that many lens spaces have infinite order in the…
A link of an isolated singularity of a two-dimensional semialgebraic surface in $R^4$ is a knot (or a link) in $S^3$. Thus the ambient Lipschitz classification of surface singularities in $R^4$ can be interpreted as a bi-Lipschitz…