Related papers: Virtual Biquandle Cocycle Quiver Representations
Biquandles and multiple conjugation biquandles are algebras which are related to links and handlebody-links in $3$-space. Cocycles of them can be used to construct state-sum type invariants of links and handlebody-links. In this paper we…
We introduce two polynomial invariants $V_1(K;t)$ and $V_2(K;t)$ of a long virtual knot $K$, which generalize the degree-two finite type invariants $v_{2,1}$ and $v_{2,2}$ of Goussarov, Polyak, and Viro. We establish their fundamental…
We define and study virtual representation spaces having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual…
We characterize pairs (Q,d) consisting of a quiver Q and a dimension vector d, such that over a given algebraically closed field k there are infinitely many representations of Q of dimension vector d. We also present an application of this…
We construct an invariant of virtual knots which is a sliceness obstruction and sensitive to the $\Delta$-move. This invariants works if $\Z_{2}\oplus \Z_{2}$-index of chords is present.
We define invariants of unoriented knots and links by enhancing the integral kei counting invariant Phi_X^Z (K) for a finite kei X using representations of the kei algebra, Z_K[X], a quotient of the quandle algebra Z[X] defined by…
In our previous papers we introduced categorical invariants, which are, roughly speaking, sets of triangulated subcategories in a given triangulated category and their quotients. Here is extended the list of examples, where these sets are…
We introduce the notion of filtered representations of quivers, which is related to usual quiver representations, but is a systematic generalization of conjugacy classes of $n\times n$ matrices to (block) upper triangular matrices up to…
A new general eigenvalue formula for the eigenvalues of Casimir invariants, for the type-I quantum superalgebras, is applied to the construction of link polynomials associated with {\em any} finite dimensional unitary irrep for these…
We introduce a modified rack algebra Z[X] for racks X with finite rack rank N. We use representations of Z[X] into rings, known as rack modules, to define enhancements of the rack counting invariant for classical and virtual knots and…
We construct a virtual quandle for links in lens spaces $L(p,q)$, with $q=1$. This invariant has two valuable advantages over an ordinary fundamental quandle for links in lens spaces: the virtual quandle is an essential invariant and the…
We introduce a new equivalence relation on decorated ribbon graphs, and show that its equivalence classes directly correspond to virtual links. We demonstrate how this correspondence can be used to convert any invariant of virtual links…
This paper is a survey on invariants of representations of quivers and their generalizations. We present the description of generating systems for invariants and relations between generators.
For any virtual link, a class of new links can be defined called stacks, in which copies of the virtual link are placed on top of one another. The resulting virtual link depends only on the virtual isotopy class of the original link, and…
Motivated by the problem of classifying quantum symmetries of non-semisimple, finite-dimensional associative algebras, we define a notion of connection between bounded quivers and build a bicategory of bounded quivers and quiver…
The theory of welded and extended welded knots is a generalization of classical knot theory. Welded (resp. extended welded) knot diagrams include virtual crossings (resp. virtual crossings and wen marks) and are equivalent under an extended…
We modify the definition of spherical knotoids to include a framing, in analogy to framed knots, and define a further modification that includes a secondary 'coframing' to obtain 'biframed' knotoids. We exhibit topological spaces whose…
We present equivalences between certain categories of vector bundles on projective varieties, namely cokernel bundles, Steiner bundles, syzygy bundles, and monads, and full subcategories of representations of certain quivers. As an…
We introduce a local deformation called the virtualized $\Delta$-move for virtual knots and links. We prove that the virtualized $\Delta$-move is an unknotting operation for virtual knots. Furthermore we give a necessary and sufficient…
We observe that any knot invariant extends to virtual knots. The isotopy classification problem for virtual knots is reduced to an algebraic problem formulated in terms of an algebra of arrow diagrams. We introduce a new notion of finite…