Detecting Whitney disks for link maps in the four-sphere

It is an open problem whether Kirk’s [Formula: see text]-invariant is the complete obstruction to a link map [Formula: see text] being link homotopic to the trivial link. The link homotopy invariant associates to such a link map [Formula: see text] a pair [Formula: see text], and we write [Formula: see text]. With the objective of constructing counterexamples, Li proposed a link homotopy invariant [Formula: see text] such that [Formula: see text] is defined on the kernel of [Formula: see text] and which also obstructs link null-homotopy. We show that, when defined, the invariant [Formula: see text] is determined by [Formula: see text], and is strictly weaker. In particular, this implies that if a link map [Formula: see text] has [Formula: see text], then after a link homotopy the self-intersections of [Formula: see text] may be equipped with framed, immersed Whitney disks in [Formula: see text] whose interiors are disjoint from [Formula: see text].

On Slavik Jablan’s work on 4-moves

We show that every alternating link of two components and [Formula: see text] crossings can be reduced by [Formula: see text]-moves to the trivial link or the Hopf link. It answers the question asked in one of the last papers by Slavik Jablan.

Plane number of links

Let L = L1 ∪ L2 ∪ ⋯ ∪ Ln be a link in ℝ3 such that Li is a trivial link for each i, 1 ≤ i ≤ n. Let P1, P2,…,Pn be mutually distinct flat planes in ℝ3 such that no two of them are parallel. Then there is a link [Formula: see text] in ℝ3 such that L is ambient isotopic to L′ and [Formula: see text] for each i, 1 ≤ i ≤ n.

Invariants of virtual rational moves

Generalized Reidemeister moves provide an extended set of moves to work with virtual knots and links. We introduce virtual tangle moves, generalization of classical rational tangle moves and show that such generalizations are essential to develop new invariants of virtual knots and links. We show that every 2-algebraic virtual link is a virtual 4-move equivalent to a trivial link or Hopf link. The properties of virtual tangle move are analyzed on few existing invariants associated with virtual knots and links.

ON 4-MOVE EQUIVALENCE CLASSES OF KNOTS AND LINKS OF TWO COMPONENTS

We study equivalence classes of knots and links of 2 components modulo 4-move. We show that all knots up to 12 crossings and knots in the family 6* reduce by 4-moves to the trivial knot. We also prove that links of 2 components with 11 crossings, and links 6* a1.a2.a3.a4.a5.a6 such that ai is a 2-algebraic tangle with no trivial components reduce to either the trivial link or to the Hopf link. For alternating links of 2-components with 12 we show that L reduces by 4-moves to either trivial link or to the Hopf link whenever L is different than 9*.2 : .2 : .2 (or its mirror image). We suggest the alternating link 9*.2 : .2 : .2 with 12 crossings as a potential example to answer the Problem 1.1(iii) in negative.

RATIONAL MOVES ON LINKS: (2, 2)-MOVES AS A CASE STUDY

We consider the problem of classification of links up to (2, 2)-moves. Our motivation comes from the theory of skein modules, more specifically from the skein module of S3 based on the deformation of (2, 2)-move. As it was proved in D-P-2, not every link can be reduced to a trivial link by (2, 2)-moves, for instance, the closure of (σ1σ2)6. In this paper, we classify 3-braids up to (2, 2)-moves and, we show how the Harikae–Nakanishi–Uchida conjecture can be modified to hold for closed 3-braids. As an important step in the classification we prove the conjecture for 2-algebraic links and classify (2, 2)-equivalence classes for links up to nine crossings. We also analyze an action of (2, 2)-move on Kei (involutive quandle) associated to a link diagram. We define Burnside Kei, Q(m, n), and ask for which values of m and n, is Q(m, n) finite. This question is motivated by classical Burnside problem.

ALMOST POSITIVE LINKS HAVE NEGATIVE SIGNATURE

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.