Narrow Results

We introduce Niebrzydowski algebras, algebraic structures with a ternary operation and a partially defined multiplication, with axioms motivated by the Reidemeister moves for $Y$-oriented trivalent spatial graphs and handlebody-links. As part of this definition, we identify generating sets of $Y$-oriented Reidemeister moves. We give some examples to demonstrate that the counting invariant can distinguish some $Y$-oriented trivalent spatial graphs and handlebody-links.

In this paper, colorings by symmetric quandles for spatial graphs and handlebody-links are introduced. We also introduce colorings by LH-quandles for LH-links. LH-links are handlebody-links, some of whose circle components are specified, which are related to Heegaard splittings of link exteriors. We also discuss quandle (co)homology groups and cocycle invariants.

In this paper, we introduce the (co)homology group of a multiple conjugation biquandle. It is the (co)homology group of the prismatic chain complex, which is related to the homology of foams introduced by J. S. Carter, modulo a certain subchain complex. We construct invariants for $S^1$-oriented handlebody-links using $2$-cocycles. When a multiple conjugation biquandle $X\times {\mathbb{Z}}_{type{X}_Y}$ is obtained from a biquandle $X$ using $n$-parallel operations, we provide a $2$-cocycle (or $3$-cocycle) of the multiple conjugation biquandle $X\times {\mathbb{Z}}_{type{X}_Y}$ from a $2$-cocycle (or $3$-cocycle) of the biquandle $X$ equipped with an $X$-set $Y$.

We introduce two notions of quandle polynomials for $G$-families of quandles: the quandle polynomial of the associated quandle and a $G$-family polynomial with coefficients in the group ring of $G$. As an application we define image subquandle polynomial enhancements of the $G$-family counting invariant for trivalent spatial graphs and handlebody-links. We provide examples to show that the new enhancements are proper.

In this paper we study isotopy classes of closed connected orientable surfaces in the standard $3$-sphere. Such a surface splits the $3$-sphere into two compact connected submanifolds, and by using their Heegaard splittings, we obtain a $2$-component handlebody-link. In this paper, we first show that the equivalence class of such a 2-component handlebody-link up to attaching trivial $1$-handles can recover the original surface. Therefore, we can reduce the study of surfaces in the $3$-sphere to that of $2$-component handlebody-links up to stabilizations. Then, by using $G$-families of quandles, we construct invariants of $2$-component handlebody-links up to attaching trivial $1$-handles, which lead to invariants of surfaces in the $3$-sphere. In order to see the effectiveness of our invariants, we will also show that our invariants can distinguish certain explicit surfaces in the $3$-sphere.