Journal of Knot Theory and Its Ramifications

An important geometric invariant of links in lens spaces is the lift in the 3-sphere of a link L in L(p, q), that is the counterimage of L under the universal covering of L(p, q). If lens spaces are defined as a lens with suitable boundary identifications, then a link in L(p, q) can be represented by a disk diagram, that is to say, a regular projection of the link on a disk. Starting from this diagram of L, we obtain a diagram of the lift in S³. Using this construction, we are able to find different knots and links in L(p, q) having equivalent lifts, that is to say, we cannot distinguish different links in lens spaces only from their lift.

If a rectangular diagram represents the trivial knot, then it can be deformed into the trivial rectangular diagram with only four edges by a finite sequence of merge operations and exchange operations, without increasing the number of edges, which was shown by Dynnikov in [Arc-presentations of links: Monotone simplification, Fund. Math. 190 (2006) 29–76; Recognition algorithms in knot theory, Uspekhi Mat. Nauk 58 (2003) 45–92. Translation in Russian Math. Surveys 58 (2003) 1093–1139]. Using this, Henrich and Kauffman gave in [Unknotting unknots, preprint (2011), arXiv:1006.4176v4 [math.GT]] an upper bound for the number of Reidemeister moves needed for unknotting a knot diagram of the trivial knot. However, exchange or merge moves on the top and bottom pairs of edges of rectangular diagrams are not considered in [Unknotting unknots, preprint (2011), arXiv:1006.4176v4 [math.GT]]. In this paper, we show that there is a rectangular diagram of the trivial knot which needs such an exchange move for being unknotted, and study upper bound of the number of Reidemeister moves needed for realizing such an exchange or merge move.

In this paper, we describe the recursion relations between two parameter HOMFLY and Kauffman polynomials of framed links. These relations correspond to embeddings of quantized universal enveloping algebras. The relation corresponding to embeddings g_n ⊃ g_k × sl_n-k where g_n is either so_2n+1, so_2n or sp_2n is new.

In this paper, we obtain new dilogarithm identities by computing the Seifert volume of manifolds obtained by Dehn surgery on the figure-eight knot in two different ways.

This paper bounds the computational cost of computing the Kauffman bracket of a link in terms of the crossing number of that link. Specifically, it is shown that the image of a tangle with g boundary points and n crossings in the Kauffman bracket skein module is a linear combination of O(2^g) basis elements, with each coefficient a polynomial with at most n non-zero terms, each with integer coefficients, and that the link can be built one crossing at a time as a sequence of tangles with maximum number of boundary points bounded by for some C. From this it follows that the computation of the Kauffman bracket of the link takes time and memory a polynomial in n times .

We apply the techniques of totally twisted Khovanov homology to Asaeda, Przytycki, and Sikora's construction of Khovanov type homologies for links and tangles in I-bundles over (orientable) surfaces. As a result we describe a chain complex built out of resolutions with only noncontractible circles whose homology is an invariant of the tangle. We use these to understand the δ-graded homology for links with alternating diagrams in the surface.

We derive explicit defining equations for a number of irreducible maximizing plane sextics with double singular points only. For most real curves, we also compute the fundamental group of the complement; all groups found are abelian, which suffices to complete the computation of the groups of all non-maximizing irreducible sextics. As a by-product, examples of Zariski pairs in the strongest possible sense are constructed.

A sufficient condition that a closed connected definite 4-manifold with infinite cyclic fundamental group is TOP-split is given. By this condition, it is shown that every closed connected definite smooth 4-manifold with infinite cyclic fundamental group is TOP-split. By combining with an earlier result, it is confirmed that every closed connected oriented smooth 4-manifold with infinite cyclic fundamental group is TOP-split. This also implies that every smooth sphere-knot in a closed simply connected smooth 4-manifold is topologically unknotted if the fundamental group of the complement is infinite cyclic.