Research
Research
Research
Research
1,808+ open-access research outputs.
We classify fillable contact structures on all negative-definite star-shaped plumbings. Along the way, we show that such Seifert fibred spaces admit a unique negative maximal twisting number, and comp…
We adapt the Ozsv\'ath-Szab\'o full path algorithm to every star-shaped graph and establish a correspondence between negative-twisting tight contact structures on any Seifert fibred space over $S^2$, …
Knots and links represent a fundamental motif of non-local connectivity that permeates the physical sciences from string theory to protein folds. While spectral braiding has been explored in two-band …
We develop a Koszul-theoretic framework for comparing classical Alexander-type invariants with infinitesimal invariants arising from finite-type commutative differential graded algebra models. The cen…
Let $Y_-$ and $Y_+$ be two compact 3-manifolds with empty or toroidal boundary. A 4-dimensional ribbon homology cobordism is a homologically trivial cobordism built with 1-handles and 2-handles. In th…
We prove a suitable fibration theorem over quasi-trivial tori that, through an approach developed by Harpaz and Wittenberg, implies so-called solvable descent. In particular, this gives a positive ans…
In this paper, we investigate a quandle structure induced by an augmented rack arising from a gauge transformation group. We construct a quandle from a principal bundle and its discrete generalization…
In this paper, we investigate structural properties of the Cayley graph of a quandle and describe this graph for several important classes of quandles, including conjugation, Takasaki, dihedral, and A…
We determine the irreducible ${\rm SL}(2,\mathbb{C})$-character variety of the 3-chain link exterior which is called the `magic $3$-manifold', and deduce a formula for the twisted Alexander polynomial…
It is well known that for $m\geq 2$ there are at most two non-equivalent $m$-knots with diffeomorphic exterior. Such pair of knots will be called $\textit{ non-reflexive knot pair}$. A classical probl…
For arborescent links, we present an efficient method of computing their Alexander polynomials. Applying this method, we express the Alexander polynomials of Montesinos links in terms of certain funct…
In this article, we investigate when the ordinary and symbolic powers of the Alexander dual of connected ideals of graphs coincide, and provide a complete classification of all such graphs. Furthermor…
We study the homological properties of $\Delta_{\mathbf{r}}(n_1, \dots, n_e)$, a simplicial complex formed by sequentially gluing complete graphs along $(r_i-1)$-simplices. This construction generates…
A perturbative expansion of knot invariants is derived using quantum cluster algebras. By interpreting the $R$-matrix of $U_q(\mathfrak{sl}_2)$ as a cluster transformation and introducing an auxiliary…
We introduce and study strongly vertex dismissible, vertex dismissible, and scalable simplicial complexes as non-pure extensions of vertex decomposability and shellability. Strong vertex dismissibilit…
In this paper, we study the embedding problem of homogeneous quandles. We give a necessary and sufficient condition under which a quandle homomorphism from the homogeneous quandle associated with a qu…
In this paper, we introduce \textit{graph-pretzel links}, a generalization of classical pretzel links based on spatial graph projections. As our main result, we investigate a subfamily associated with…
We present a complete classification of spherical knotoids with up to six crossings and conjecture that our classification up to seven crossings is complete. Our work extends the tradition of knot tab…
We present explicit infinite families of twisted torus knots that are not fibered. Our approach relies on an explicit formula for the Alexander polynomial derived in our previous work. We show that th…
We prove a sharp inequality between the Alexander-Taylor capacity and the functional capacity in a complex Sobolev space on a compact K\"ahler manifold. The latter space and capacity were introduced b…
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