Related papers: When is a band-connected sum equal to the connecte…

We show that if a composite $\theta$-curve has (proper rational) unknotting number one, then it is the order 2 sum of a (proper rational) unknotting number one knot and a trivial $\theta$-curve. We also prove similar results for 2-strand…

Let $\Sigma_0,\Sigma_1$ be closed oriented surfaces. Two oriented knots $K_0 \subset \Sigma_0 \times [0,1]$ and $K_1 \subset \Sigma_1 \times [0,1]$ are said to be (virtually) concordant if there is a compact oriented $3$-manifold $W$ and a…

A $k$-sum of a set $A\subseteq \mathbb{Z}$ is an integer that may be expressed as a sum of $k$ distinct elements of $A$. How large can the ratio of the number of $(k+1)$-sums to the number of $k$-sums be? Writing $k\wedge A$ for the set of…

We show that there exist infinitely many examples of pairs of knots, K_1 and K_2, that have no epimorphism $\pi_1(S^3\setminus K_1) \to \pi_1(S^3\setminus K_2)$ preserving peripheral structure although their A-polynomials have the…

Let n be a positive integer. We provide a Khovanov homology proof of the following classical fact: If the closure of an n-strand braid is the n-component unlink, then the braid is trivial.

We study a certain type of braid closure which resembles the plat closure but has certain advantages; for example, it maps pure braids to knots. The main results of this note are a Markov-type theorem and a description of how Vassiliev…

For $b>1$ and $\alpha\beta$ a string of two digits in base $b$, let $K_1$ be the subsum of the harmonic series with only those integers having exactly one occurrence of $\alpha\beta$. We obtain a theoretical representation of such $K_1$…

Modular operads relevant to string theory can be equipped with an additional structure, coming from the connected sum of surfaces. Motivated by this example, we introduce a notion of connected sum for general modular operads. We show that a…

Let $K$ be a knot type for which the quadratic term of the Conway polynomial is nontrivial, and let $\gamma: \mathbb{R}\to \mathbb{R}^3$ be an analytic $\mathbb{Z}$-periodic function with non-vanishing derivative which parameterizes a knot…

Here, we prove that $0-$shake slice knots are slice.

We define direct sums and a corresponding notion of connectedness for graph limits. Every graph limit has a unique decomposition as a direct sum of connected components. As is well-known, graph limits may be represented by symmetric…

Let $C_b(X)$ be the C*-algebra of bounded continuous functions on some non-compact, but locally compact Hausdorff space $X$. Moreover, let $A_0$ be some ideal and $A_1$ be some unital C*-subalgebra of $C_b(X)$. For $A_0$ and $A_1$ having…

Ribbon concordances between knots generalize the notion of ribbon knots. Agol, building on work of Gordon, proved ribbon concordance gives a partial order on knots in $S^3$. In previous work, the author and Greene conjectured that positive…

We give a complete classification of links up to clasp-pass moves, which coincides with Habiro's $C_3$-equivalence. We also classify links up to band-pass and band-# moves, which are versions of the usual pass- and #-move, respectively,…

We study the double slice genus of a knot, a natural generalization of slice genus. We define a notion called band number, a natural generalization of band unknotting number, and prove it is an upper bound on double slice genus. Our bound…

We use an example to provide evidence for the statement: the Vassiliev-Kontsevich invariants $k_n$ of a knot (or braid) $k$ can be redefined so that $k = \sum_0^\infty k_n$. This constructs a knot from its Vassiliev-Kontsevich invariants,…

This paper investigates the relationship between the signature and the crossing number of knots and links. We refine existing theorems and provide a comprehensive classification of links with specific properties, particularly those with…

A surface-link is a closed surface embedded in the 4-space, possibly disconnected or non-orientable. Every surface-link can be presented by the plat closure of a braided surface, which we call a plat form presentation. The knot symmetric…

Let C(n,k) denote the number of connected graphs with n labeled vertices and n+k-1 edges. For any sequence (k_n), the limit of C(n,k_n) as n tends to infinity is known. It has been observed that, if k_n=o(\sqrt{n}), this limit is…

For integers $k \geq 2$ and $n \geq k+1$, we prove the following: If $n\cdot k$ is even, there is a connected $k$-regular graph on $n$ vertices. If $n\cdot k$ is odd, there is a connected nearly $k$-regular graph on $n$ vertices.