Related papers: A counterexample to the Bernhard-Jablan unknotting…
We study symmetric crossing change operations for strongly invertible knots. Our main theorem is that the most natural notion of equivariant unknotting number is not additive under connected sum, in contrast with the longstanding conjecture…
A bridge position of a knot is said to be perturbed if there exists a cancelling pair of bridge disks. Motivated by the examples of knots admitting unperturbed strongly irreducible non-minimal bridge positions due to…
We prove that a special alternating knot does not decompose as a non-trivial band sum. This restricts concordances from special alternating knots, and we conjecture that special alternating knots are ribbon concordance minimal. We verify…
For any link in the 3-sphere, there is a natural lower bound for the unlinking number in terms of the classical signature. We prove that if this lower bound is sharp for a special alternating link $L$, then the unlinking number of $L$ is…
An $n$-crossing is a point in the projection of a knot where $n$ strands cross so that each strand bisects the crossing. An \"ubercrossing projection has a single $n$-crossing and a petal projection has a single $n$-crossing such that there…
Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by V.~Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter…
We consider diagrams of links in $S^2$ obtained by projection from $S^3$ with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots…
Using unknotting number, we introduce a link diagram invariant of Hass and Nowik type, which changes at most by 2 under a Reidemeister move. As an application, we show that a certain infinite sequence of diagrams of the trivial…
We study decomposition into simple arcs (i. e., arcs without self-intersections) for diagrams of knots and spatial graphs. In this paper, it is proved in particular that if no edge of a finite spatial graph $G$ is a knotted loop, then there…
Introducing a way to modify knots using $n$-trivial rational tangles, we show that knots with given values of Vassiliev invariants of bounded degree can have arbitrary unknotting number (extending a recent result of Ohyama, Taniyama and…
We extend techniques due to Pardon to show that there is a lower bound on the distortion of a knot in $\mathbb{R}^3$ proportional to the minimum of the bridge distance and the bridge number of the knot. We also exhibit an infinite family of…
We prove that a crossing change along a double point circle on a 2-knot is realized by ribbon-moves for a knotted torus obtained from the 2-knot by attaching a 1-handle. It follows that any 2-knots for which the crossing change is an…
It is known that the arc index of alternating knots is the minimal crossing number plus two and the arc index of prime nonalternating knots is less than or equal to the minimal crossing number. We study some cases when the arc index is…
The connected sum of two flat virtual knots depends on the choice of diagrams and basepoints. We show that any minimal crossing diagram of a composite flat virtual knot is a connected sum diagram. We also show the crossing number of flat…
The number of colorings of a knot diagram by a quandle has been shown to be a knot invariant by CJKLS using quandle cohomology methods. In a previous paper by the second named author, the CJKLS invariant was refined and, in particular, it…
We study the gordian graph of all knots in $\R^3$: two knots are adjacent if they differ by a single crossing change. We prove that this graph contains isometrically an infinite countable tree with infinite valency, and that the complement…
In [3] we constructed the parity-biquandle bracket valued in {\em pictures} (linear combinations of $4$-valent graphs). We gave no example of classical links such that the parity-biquandle bracket of which is not trivial. In the present…
We introduce an unknotting-type number of knot projections that gives an upper bound of the crosscap number of knots. We determine the set of knot projections with the unknotting-type number at most two, and this result implies classical…
We show that every knot is one crossing change away from a knot of arbitrarily high bridge number and arbitrarily high bridge distance.
We consider knot theories possessing a {\em parity}: each crossing is decreed {\em odd} or {\em even} according to some universal rule. If this rule satisfies some simple axioms concerning the behaviour under Reidemeister moves, this leads…