相关论文: Uniqueness of higher genus bridge surfaces for tor…

Suppose M is a closed irreducible orientable 3-manifold, K is a knot in M, P and Q are bridge surfaces for K and K is not removable with respect to Q. We show that either Q is equivalent to P or $d(K,P) \leq 2-\chi(Q-K)$. If K is not a two…

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…

We show that no torus knot of type $(2,n)$, $n>3$ odd, can be obtained from a polynomial embedding $t \mapsto (f(t), g(t), h(t))$ where $(\deg(f),\deg(g))\leq (3,n+1) $. Eventually, we give explicit examples with minimal lexicographic…

We determine the Dehn surgeries on 2-bridge links, which yield reducible 3-manifolds. Further, we show the conditions that we obtain a torus or cable knot from one component of a 2-bridge link by a surgery on another component.

We prove that the meridional rank and the bridge number of the Whitehead double of a prime algebraically tame knot coincide. Algebraically tame knots are a broad generalization of torus knots and iterated cable knots.

We show that a hyperbolic 2-bridge knot complement is the unique knot complement in its commensurability class. We also discuss constructions of commensurable hyperbolic knot complements and put forth a conjecture on the number of…

The non-orientable 4-genus of a knot $K$ in $S^{3}$, denoted $\gamma_4(K)$, measures the minimum genus of a non-orientable surface in $B^{4}$ bounded by $K$. We compute bounds for the non-orientable 4-genus of knots $T_{5, q}$ and $T_{6,…

We exhibit pairs of transverse knots with the same self-linking number that are not transversely isotopic, using the recently defined knot Floer homology invariant for transverse knots and some algebraic refinements of it.

We consider a surface link in the 4-space which can be presented by a simple branched covering over the standard torus, which we call a torus-covering link. Torus-covering links include spun $T^2$-knots and turned spun $T^2$-knots. In this…

We show that every non-trivial tame knot or link in R^3 has a quadrisecant, i.e. four collinear points. The quadrisecant must be topologically non-trivial in a precise sense. As an application, we show that a nonsingular, algebraic surface…

We show that the $(4,5)$- and $(5,6)$-torus knots admit ghost characters. Consequently, these knots provide counterexamples to Ng's conjecture, which proposes an isomorphism between the complexification of degree $0$ abelian knot contact…

This paper concerns the H(2)-unknotting numbers of links related to 2-bridge links. It consists of three parts. In the first part, we consider a necessary and sufficient condition for a 2-bridge link to have H(2)-unknotting number one. The…

A knot K is called n-adjacent to the unknot, if K admits a projection containing n generalized crossings such that changing any m (no larger than n) of them yields a projection of the unknot. We show that a non-trivial satellite knot K is…

If R is a nonseparating simple closed curve on the boundary of a genus two handlebody H and H[R] has incompressible boundary, then there exists a unique arc omega in bdry(H), meeting R only in its endpoints, such that, omega is isotopic in…

We classify all knot diagrams of genus two and three, and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof…

We consider the relations $\ge$ and $\ge_p$ on the collection of all knots, where $k \ge k'$ (respectively, $k \ge_p k'$) if there exists an epimorphism $\pi k \to \pi k'$ of knot groups (respectively, preserving peripheral systems). When…

A theorem of Jorgensen and Thurston implies that the volume of a hyperbolic 3-manifold is bounded below by a linear function of its Heegaard genus. Heegaard surfaces and bridge surfaces often exhibit similar topological behavior; thus it is…

We show that twisted torus knots $T(p,q,3,s)$ are tunnel number one. A short spanning arc connecting two adjacent twisted strands is an unknotting tunnel.

The genus non-increasing totally positive unknotting number is the minimum number of crossing changes that transform a knot into the unknot, such that all the crossing changes are positive-to-negative crossing changes that do not increase…

We give examples of a linear combination of algebraic knots and their mirrors that are algebraically slice, but whose topological and smooth four-genus is two. Our examples generalize an example of non-slice algebraically slice linear…