Narrow Results

We study the existence of relations between the degrees of the knot polynomials and some classical knot invariants, partially confirming and extending the question of Morton on the skein polynomial and a recent question of Ferrand.

We compute the slice euler characteristic of certain links by using the so-called generalized adjunction formula, which is proved by Kronheimer, Mrowka, Morgan, Szabó and Taubes. Furthermore, for links obtained from such links by band surgery, we estimate the unknotting numbers, the 4-dimensional clasp numbers, and the slice euler characteristic.

S-equivalence of classical knots is investigated, as well as its relationship with mutation and the unknotting number. Furthermore, we identify the kernel of Bredon's double suspension map, and give a geometric relation between slice and algebraically slice knots. Finally, we show that every knot is S-equivalent to a prime knot.

In this note, we will give an approach to determine the unknotting number by a surgical view of Alexander matrix.

Computing unlinking number is usually very difficult and complex problem, therefore we define BJ-unlinking number and recall Bernhard–Jablan conjecture stating that the classical unknotting/unlinking number is equal to the BJ-unlinking number. We compute BJ-unlinking number for various families of knots and links for which the unlinking number is unknown. Furthermore, we define BJ-unlinking gap and construct examples of links with arbitrarily large BJ-unlinking gap. Experimental results for BJ-unlinking gap of rational links up to 16 crossings, and all alternating links up to 12 crossings are obtained using programs LinKnot and K2K. Moreover, we propose families of rational links with arbitrarily large BJ-unlinking gap and polyhedral links with constant non-trivial BJ-unlinking gap. Computational results suggest existence of families of non-alternating links with arbitrarily large BJ-unlinking gap.

In this paper, we show that unknotting number of the connected sum of n identical knots k is at least n when k has nontrivial Alexander polynomial.

In this paper we define a partial ordering of knots and links using a special property derived from their minimal diagrams. A link is called a predecessor of a link if and a diagram of can be obtained from a minimal diagram D of by a single crossing change. In such a case, we say that . We investigate the sets of links that can be obtained by single crossing changes over all minimal diagrams of a given link. We show that these sets are specific for different links and permit partial ordering of all links. Some interesting results are presented and many questions are raised.

We show that for any nontrivial knot K and any natural number n, there is a diagram D of K such that the unknotting number of D is greater than or equal to n. It is well-known that twice the unknotting number of K is less than or equal to the crossing number of K minus one. We show that the equality holds only when K is a (2, p)-torus knot.

We introduce a new numerical invariant of knots and links from the descending diagrams. It is considered to live between the unknotting number and the bridge number. Some fundamental results and an incomplete table of the invariant for knots with 8-crossings or less are given.

We analyze properties of links which have diagrams with a small number of negative crossings. We show that if a nontrivial link has a diagram with all crossings positive except possibly one, then the signature of the link is negative. If a link diagram has two negative crossings, we show that the signature of the link is nonpositive with the exception of the left-handed Hopf link (possibly, with extra trivial components). We also characterize those links which have signature zero and diagrams with two negative crossings. In particular, we show that if a nontrivial knot has a diagram with two negative crossings then the signature of the knot is negative, unless the knot is a twist knot with negative clasp. We completely determine all trivial link diagrams with two or fewer negative crossings. For a knot diagram with three negative crossings, the signature of the knot is nonpositive except for the left-handed trefoil knot. These results generalize those of Rudolph, Cochran, Gompf, Traczyk and Przytycki, solve [27, Conjecture 5], and give a partial answer to [3, Problem 2.8] about knots dominating the trefoil knot or the trivial knot. We also describe all unknotting number one positive knots.

It is well-known that for any link L, twice the unknotting number of L is less than or equal to the crossing number of L. Taniyama characterized the links which satisfy the equality. We characterize the links where twice the unknotting number is equal to the crossing number minus one. As a corollary, we show that for any link L with twice the unknotting number of L is greater than or equal to the crossing number of L minus two, every minimal diagram of L realizes the unknotting number.

A pseudodiagram is a diagram of a knot with some crossing information missing. We review and expand the theory of pseudodiagrams introduced by Hanaki. We then extend this theory to the realm of virtual knots, a generalization of knots. In particular, we analyze the trivializing number of a pseudodiagram, i.e. the minimum number of crossings that must be resolved to produce the unknot. We consider how much crossing information is needed in a virtual pseudodiagram to identify a non-trivial knot, a classical knot, or a non-classical knot. We then apply pseudodiagram theory to develop new upper bounds on unknotting number, virtual unknotting number, and genus.

In this paper, we give a bound for the Δ-unknotting number of a Whitehead double in terms of the unknotting number and a certain integral invariant of its companion knot. As applications, we show that the Δ-unknotting number of m-twisted Whitehead doubles of certain knots does not remember its companion knot, and is equal to the twist number m. We also give possible Δ-unknotting number of m-twisted Whitehead doubles whose companions are knots with unknotting number 1, certain twist knots, amphicheiral knots, and positive knots.

It is known that a knot/link can be nullified, i.e. can be made into the trivial knot/link, by smoothing some crossings in a projection diagram of the knot/link. The nullification of knots/links is believed to be biologically relevant. For example, in DNA topology, the nullification process may be the pathway for a knotted circular DNA to unknot itself (through recombination of its DNA strands). The minimum number of such crossings to be smoothed in order to nullify the knot/link is called the nullification number. It turns out that there are several different ways to define such a number, since different conditions may be applied in the nullification process. We show that these definitions are not equivalent, thus they lead to different nullification numbers for a knot/link in general, not just one single nullification number. Our aim is to explore some mathematical properties of these nullification numbers. First, we give specific examples to show that the nullification numbers we defined are different. We provide a detailed analysis of the nullification numbers for the well known 2-bridge knots and links. We also explore the relationships among the three nullification numbers, as well as their relationships with other knot invariants. Finally, we study a special class of links, namely those links whose general nullification number equals one. We show that such links exist in abundance. In fact, the number of such links with crossing number less than or equal to n grows exponentially with respect to n.

It is known that any virtual knot can be deformed into the trivial knot by a finite sequence of forbidden moves. In this paper, we give the difference of the values obtained from some invariants constructed by Henrich between two virtual knots which can be transformed into each other by a single forbidden move. As a result, we obtain a lower bound of the unknotting number of a virtual knot by forbidden moves.

It is known that any surface knot can be transformed to an unknotted surface knot or a surface knot which has a diagram with no triple points by a finite number of 1-handle additions. The minimum number of such 1-handles is called the unknotting number or the triple point cancelling number, respectively. In this paper, we give upper bounds and lower bounds of unknotting numbers and triple point cancelling numbers of torus-covering knots, which are surface knots in the form of coverings over the standard torus T. Upper bounds are given by using m-charts on T presenting torus-covering knots, and lower bounds are given by using quandle colorings and quandle cocycle invariants.

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) ≤ g(K) holds, and that if u(K) = g(K) then K is either a 2-braid knot, a connected sum of two 2-braid knots, the figure-eight knot, a strongly quasipositive knot or its mirror image.

We list prime knots with up to 12 crossings having the property: the lower bound of the unknotting number cannot be decided by the signature or non-triviality but is given by either: (i) the condition of a 2-bridge knot with unknotting number one, (ii) the criteria using special values of the Jones, Q, or HOMFLYPT polynomials, or (iii) Wendt's formula. Then we can give new information to the table of unknotting numbers in "KnotInfo", a web-based table of knot invariants.

In [Non-reidemeister knot theory and its applications in dynamical systems, geometry, and topology, preprint (2015), arXiv:1501.05208.] the first author gave the definition of $k$-free braid groups $G_n^k$. Here we establish connections between free braid groups, classical braid groups and free groups: we describe explicitly the homomorphism from (pure) braid group to $k$-free braid groups for important cases $k=3,4$. On the other hand, we construct a homomorphism from (a subgroup of) free braid groups to free groups. The relations established would allow one to construct new invariants of braids and to define new powerful and easily calculated complexities for classical braid groups.

A crossing change of a handlebody-knot is that of a spatial graph representing it. We see that any handlebody-knot can be deformed into trivial one by some crossing changes. So we define the unknotting numbers for handlebody-knots. In the case classical knots, which are considered as genus one handlebody-knots, Clark, Elhamdadi, Saito and Yeatman gave lower bounds of the Nakanishi indices by the numbers of some finite Alexander quandle colorings, and hence they also gave lower bounds of the unknotting numbers. In this paper, we give lower bounds of the unknotting numbers for handlebody-knots with any genus by the numbers of some finite Alexander quandle colorings of type at most $3$.