Related papers: Tangle solutions for composite knots: applications…
In this paper we study further when tangles embed into the unknot, the unlink or a split link. In particular, we study obstructions to these properties through geometric characterizations, tangle sums and colorings. As an application we…
We refine the combinatorial 1-cocycle $\mathbb{L}R_{reg}$ for regular isotopies of long knots to a 1-cocycle with values in the free $\mathbb{Z}[x,x^{-1}]$-module generated by regular isotopy classes of oriented tangles with exactly one…
We define composite DAHA-superpolynomials of torus knots, depending on pairs of Young diagrams and generalizing the composite HOMFLY-PT polynomials in the theory of the skein of the annulus. We provide various examples. Our superpolynomials…
We construct an infinite collection of knots with the property that any knot in this family has $n$-string essential tangle decompositions for arbitrarily high $n$.
In this paper, we derive a general formula of the tangle for pure states of three qubits, and present three explicit local unitary (LU) polynomial invariants. Our result goes beyond the classical work of tangle, 3-tangle and von Neumann…
The affine Hecke algebra $\dot H_n$ of type $A$ is often presented as a quotient of the braid algebra of $n$-braids in the annulus. This leads to diagrammatic representations in terms of braids in the annulus, subject to a quadratic…
Given a 4-tangle shadow, we concatenate it with itself n times and form a knot by applying a closure operation that connects each top endpoint to the corresponding bottom endpoint on the same side without introducing any crossings. We then…
We develop a model characterizing all possible knots and links arising from recombination starting with a twist knot substrate, extending previous work of Buck and Flapan. We show that all knot or link products fall into three…
We rewrite the recently proposed differential expansion formula for HOMFLY polynomials of the knot $4_1$ in arbitrary rectangular representation $R=[r^s]$ as a sum over all Young sub-diagrams $\lambda$ of $R$ with extraordinary simple…
We prove that there exists no a priori bound on the Euler characteristic of a closed symplectic 4-manifold coming solely from the genus of a compatible Lefschetz pencil on it, nor is there a similar bound for Stein fillings of a contact…
We show that the Charney--Davis strict hyperbolization procedure can preserve stable tangent bundles, answering a question of Charney and Davis. The key input is the construction of many hyperbolizing pieces, obtained using separability…
A new lattice model is presented for correlated electrons on the unrestricted $4^L$-dimensional electronic Hilbert space $\otimes_{n=1}^L{\bf C}^4$ (where $L$ is the lattice length). It is a supersymmetric generalization of the Hubbard…
We give examples of compact symplectic manifolds with disconnected contact type boundary in dimension $4n$ for any $n\geq 1$. The example is given by a subset of the tangent bundle of a compact quotient of the complex hyperbolic space…
A general method in constructing a complete set of wave functions for multipartite identical qubits is presented based on the irreducible representations of the permutation group and the nth rank tensors. Particular examples for n =2, 3,…
Counterterms that are not reducible to $\zeta_{n}$ are generated by ${}_3F_2$ hypergeometric series arising from diagrams for which triangle and uniqueness relations furnish insufficient data. Irreducible double sums, corresponding to the…
We give an upper bound on the z-degree of the Kauffman polynomial of a link, using bridges of length greater than one which are separated in some tangle decomposition of a link diagram. We construct some examples by wiring together rational…
We prove that knots obtained by attaching a band to a split link satisfy the cabling conjecture. We also give new proofs that unknotting number one knots are prime and that genus is superadditive under band sum. Additionally, we prove a…
Using Monte Carlo simulations and advanced knot localization methods, we analyze the length and distribution of prime components in composite knots tied on Freely Jointed Rings (FJRs). For increasing contour length, we observe the…
Generalizing prior work of Levine, we give infinitely many examples of pattern knots P such that P(K) is not slice in any rational homology 4-ball, for any companion knot K. To show this, we establish a closed formula for the concordance…
We show that 4-connected plane triangulations can be redrawn such that edges are represented by straight segments and the vertices are covered by a set of at most $\sqrt{2n}$ lines each of them horizontal or vertical. The same holds for all…