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Geometric interpretations of some virtual knot invariants are given in terms of invariants of links in $\mathbb{S}^3$. Alexander polynomials of almost classical knots are shown to be specializations of the multi-variable Alexander…
We give a number theoretic proof of the integrality of certain BPS invariants of knots. The formulas for these numbers are sums involving binomial coefficients and the M\"obius function. We also prove a conjecture about further divisibility…
Among the knots that are the connected sum of two torus knots with cobordism distance 1, we characterize those that have 4-dimensional clasp number at least 2, and we show that their n-fold connected self-sum has 4-dimensional clasp number…
The knot Floer order $\operatorname{Ord}(K)$ is a knot invariant derived from knot Floer homology that provides bounds on many other invariants, such as the bridge index $\operatorname{br}(K)$ for which $\operatorname{Ord}(K) + 1 \leq…
The fusion number of a ribbon knot is the minimal number of 1-handles needed to construct a ribbon disk. The strong homotopy fusion number of a ribbon knot is the minimal number of 2-handles in a handle decomposition of a ribbon disk…
We solve the Jones conjecture, which states that the exponent sum in a minimal braid representation of a knot in S^3 is a knot invariant, by proving a generalized version of the original one. We apply contact geometry to study this problem…
We consider arrow diagrams of links in $S^3$ and define $k$-moves on such diagrams, for any $k\in\mathbb N$. We study the equivalence classes of links in $S^3$ up to $k$-moves. For $k=2$, we show that any two knots are equivalent, whereas…
We investigate the behaviour of Rasmussen's invariant $s$ under the sharp operation on knots and obtain a lower bound for the sharp unknotting number. This bound leads us to an interesting move that transforms arbitrary knots into…
The set of isotopy classes of nontrivial torus knots $T(p,q)$ in $S^3$ is in bijection with the set of coprime integer pairs $(p,q)$ satisfying $|p|>q\geq 2$. We verify the AJ conjecture for the connected sums $T(p,q)\# T(a,b)$ when $p$ and…
We introduce and explore the relation between knot invariants and quiver representation theory, which follows from the identification of quiver quantum mechanics in D-brane systems representing knots. We identify various structural…
Closed 3-string braids admit many bandings to two-bridge links. By way of the Montesinos Trick, this allows us to construct infinite families of knots in the connected sum of lens spaces L(r,1) # L(s,1) that admit a surgery to a lens space…
When two boundary-parabolic representations of knot groups are given, we introduce the connected sum of these representations and show several natural properties including the unique factorization property. Furthermore, the complex volume…
We construct a graph G such that any embedding of G into R^{3} contains a nonsplit link of two components, where at least one of the components is a nontrivial knot. Further, for any m < n we produce a graph H so that every embedding of H…
Birman-Menasco proved that there are finitely many knots having a given genus and braid index. We give a quantitative version of Birman-Menasco finiteness theorem, an estimate of the crossing number of knots in terms of genus and braid…
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…
The contents of this 6-page paper have been subsumed into the 13-page paper, "A note on closed 3-braids", arXiv:0802.1072 [math.GT]. This paper is correct, but contains less information than the new one. The topological classification of…
Let $G$ be a nonabelian, simple group with a nontrivial conjugacy class $C \subseteq G$. Let $K$ be a diagram of an oriented knot in $S^3$, thought of as computational input. We show that for each such $G$ and $C$, the problem of counting…
A virtual knot, which is one of generalizations of knots in $\mathbb{R}^{3}$ (or $S^{3}$), is, roughly speaking, an embedded circle in thickened surface $S_{g} \times I$. In this talk we will discuss about knots in 3 dimensional $S_{g}…
We define homotopy-theoretic invariants of knots in prime 3-manifolds. Fix a knot J in a prime 3-manifold M. Call a knot K in M concordant to J if it cobounds a properly embedded annulus with J in MxI, and call K J-characteristic if there…
We refine the Polyak-Viro Gauss diagram formula for the Vassiliev invariant of order two in a very simple way for the 2-cable of a framed long knot. Surprisingly, the resulting isotopy invariant of framed knots can detect already the…