Related papers: The unknotting number and classical invariants II
We give the first examples of a pair of knots $K_1$,$K_2$ in the 3-sphere for which their unknotting numbers satisfy $u(K_1\#K_2)<u(K_1)+u(K_2)$ . This answers question 1.69(B) from Kirby's problem list, "Problems in low-dimensional…
We show that the L^2-torsion and the von Neumann rho-invariant give rise to commensurability invariants of knots.
We present a topological interpretation of knot and braid contact homology in degree zero, in terms of cords and skein relations. This interpretation allows us to extend the knot invariant to embedded graphs and higher-dimensional knots. We…
We state Bennequin inequalities in the relative case, and show that the relative invariants are additive under relative connected sums. We show they exhibit similar limitations as their classical analogues. We study relatively Legendrian…
One construction of the Alexander polynomial is as a quantum invariant associated with representations of restricted quantum $\mathfrak{sl}_2$ at a fourth root of unity. We generalize this construction to define a link invariant…
It is shown that Legendrian (resp. transverse) cable links in the 3-sphere with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the…
We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by…
We prove that if an alternating knot has unknotting number one, then there exists an unknotting crossing in any alternating diagram. This is done by showing that the obstruction to unknotting number one developed by Greene in his work on…
We address the primary decomposition of the knot concordance group in terms of the solvable filtration and higher-order von Neumann $\rho$-invariants by Cochran, Orr, and Teichner. We show that for a nonnegative integer n, if the connected…
A polynomial is presented that models a topological knot in a unique manner. It distinguishes all types of knots including the orientation and has a group theory interpretation. The topologies may be labeled via a number, which upon a base…
We construct an invariant of virtual knots which is a sliceness obstruction and sensitive to the $\Delta$-move. This invariants works if $\Z_{2}\oplus \Z_{2}$-index of chords is present.
We use the contact invariant defined in [2] to construct a new invariant of Legendrian knots in Kronheimer and Mrowka's monopole knot homology theory (KHM), following a prescription of Stipsicz and V\'ertesi. Our Legendrian invariant…
We define an SFT-type invariant for Legendrian knots in the standard contact $\mathbb{R}^3$. The invariant is a deformation of the Chekanov-Eliashberg differential graded algebra. The differential consists of a part that counts index zero…
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…
In this paper we associate an invariant to a biquaternion algebra $B$ over a field $K$ with a subfield $F$ such that $K/F$ is a quadratic separable extension and $\operatorname{char}(F)=2$. We show that this invariant is trivial exactly…
We extend the quandle cocycle invariant to the context of stuck links. More precisely, we define an invariant of stuck links by assigning Boltzmann weights at both classical and stuck crossings. As an application, we define a…
This paper defines a new invariant of virtual knots and links that we call the extended bracket polynomial, and denote by <<K>> for a virtual knot or link K. This invariant is a state summation over bracket states of the oriented diagram…
We give empirical evidence that the UV-divergences of a renormalizable field theory are knot invariants.
We calculate the knot invariant coming from the Teichm\"{u}ller TQFT [AK1]. Specifically we calculate the knot invariant for the complement of the knot $6_1$ both in the original [AK1] and the new formulation of the Teichm\"{u}ller TQFT…
Let $M$ be a connected, closed, oriented three-manifold and $K$, $L$ two rationally null-homologous oriented simple closed curves in $M$. We give an explicit algorithm for computing the linking number between $K$ and $L$ in terms of a…