Related papers: Concordance invariants of doubled knots and blowin…

Kronheimer and Mrowka introduced a new knot invariant, called $s^\sharp$, which is a gauge theoretic analogue of Rasmussen's $s$ invariant. In this article, we compute Kronheimer and Mrowka's invariant for some classes of knots, including…

Ito-Takimura recently defined a splice-unknotting number $u^-(D)$ for knot diagrams. They proved that this number provides an upper bound for the crosscap number of any prime knot, asking whether equality holds in the alternating case. We…

We prove that for any winding number $m>0$ pattern $P$ and winding number $-m$ pattern $Q$, there exist knots $K$ such that the minimal genus of a cobordism between $P(K)$ and $Q(K)$ is arbitrarily large. This answers a question posed by…

We establish a version of Seiberg--Witten Floer $K$-theory for knots, as well as a version of Seiberg-Witten Floer $K$-theory for 3-manifolds with involution. The main theorems are 10/8-type inequalities for knots and for involutions. The…

The slicing degree of a knot $K$ is defined as the smallest integer $k$ such that $K$ is $k$-slice in $\#^n \overline{\mathbb{CP}^2}$ for some $n$. In this paper, we establish bounds for the slicing degrees of knots using Rasmussen's…

The unknotting number is the classical invariant of a knot. However, its determination is difficult in general. To obtain the unknotting number from definition one has to investigate all possible diagrams of the knot. We tried to show the…

We discuss an infinite class of metabelian Von Neumann rho-invariants. Each one is a homomorphism from the monoid of knots to the real line. In general they are not well defined on the concordance group. Nonetheless, we show that they pass…

Unknotting numbers for torus knots and links are well known. In this paper, we present a method for determining the position of unknotting number crossing changes in a toric braid B(p, q) such that the closure of the resultant braid is…

We will strengthen the known upper and lower bounds on the delta-crossing number of knots in therms of the triple-crossing number. The latter bound turns out to be strong enough to obtain (unknown values of) triple-crossing numbers for a…

A slice-torus invariant is an $\mathbb{R}$-valued homomorphism on the knot concordance group whose value gives a lower bound for the 4-genus such that the equality holds for any positive torus knot. Such invariants have been discovered in…

We prove that if an alternating 3-braid knot has unknotting number one, then there must exist an unknotting crossing in any alternating diagram of it, and we enumerate such knots. The argument combines the obstruction to unknotting number…

The theory of tunnel number 1 knots detailed in our previous paper, The tree of knot tunnels, provides a non-negative integer invariant called the depth of the tunnel. We give various results related to the depth invariant. Noting that it…

A new type of knot energy is presented via real life experiments involving a thin resilient metallic tube. Knotted in different ways, the device mechanically acquires a uniquely determined (up to isometry) normal form at least when the…

We compute the involutive concordance invariants for the 10- and 11-crossing (1,1)-knots.

Let $u(K)$ and $g(K)$ denote the unknotting number and the genus of a knot $K$, respectively. For a 3-braid knot $K$, we show that $u(K)\le g(K)$ holds, and that if $u(K)=g(K)$ then $K$ is either a 2-braid knot, a connected sum of two…

It was previously shown by the second author that every knot in $S^3$ is ambient isotopic to one component of a two-component, alternating, hyperbolic link. In this paper, we define the alternating volume of a knot $K$ to be the minimum…

A (t,s)-rack is a rack structure defined on a module over the ring $\ddot\Lambda=\mathbb{Z}[t^{\pm 1},s]/(s^2-(1-t)s)$. We identify necessary and sufficient conditions for two $(t,s)$-racks to be isomorphic. We define enhancements of the…

Knots and links in 3-manifolds are studied by applying intersection invariants to singular concordances. The resulting link invariants generalize the Arf invariant, the mod 2 Sato-Levine invariants, and Milnor's triple linking numbers.…

Let $K$ be a genus $g$ alternating knot with Alexander polynomial $\Delta_K(T)=\sum_{i=-g}^ga_iT^i$. We show that if $|a_g|=|a_{g-1}|$, then $K$ is the torus knot $T_{2g+1,\pm2}$. This is a special case of the Fox Trapezoidal Conjecture.…

We show that every two-bridge knot $K$ of crossing number $N$ admits a polynomial parametrization $x=T_3(t), y = T_b(t), z =C(t)$ where $T_k(t)$ are the Chebyshev polynomials and $b+\deg C = 3N$. If $C (t)= T_c(t)$ is a Chebyshev…