Symmetric knots and billiard knots

Symmetry of geometrical figures is reflected in regularities of their algebraic invariants. Algebraic regularities are often preserved when the geometrical figure is topologically deformed. The most natural, intuitively simple but mathematically complicated, topological objects are Knots. We present in this papers several examples, both old and new, of regularity of algebraic invariants of knots. Our main invariants are the Jones polynomial (1984) and its generalizations. In the first section, we discuss the concept of a symmetric knot, and give one important example -- a torus knot. In the second section, we give review of the Jones type invariants. In the third section, we gently and precisely develop the periodicity criteria from the Kauffman bracket (ingenious version of the Jones polynomial). In the fourth section, we extend the criteria to skein (Homflypt) and Kauffman polynomials. In the fifth section we describe r^q periodicity criteria using Vassiliev-Gusarov invariants. We also show how the skein method may be used for r^q periodicity criteria for the classical (1928) Alexander polynomial. In the sixth section, we introduce the notion of Lissajous and billiard knots and show how symmetry principles can be applied to these geometric knots. Finally, in the seventh section, we show how symmetry can be used to gain nontrivial information about knots in other 3-manifolds, and how symmetry of 3-manifolds is reflected in manifold invariants. In particular, we find the formula for a torus knot in a solid torus and we show that the third eigenvector (e_3) of the Kauffman bracket skein module of the solid torus (with respect to the meridian Dehn twist) can be realized by a link in a solid torus.Comment: 41 page, 31 figure

Q-polynomial invariant of rooted trees

We describe in this note a new invariant of rooted trees. We argue that the invariant is interesting on it own, and that it has connections to knot theory and homological algebra. However, the real reason that we propose this invariant to readers is that we deal here with an elementary, interesting, new mathematics, and after reading this essay readers can take part in developing the topic, inventing new results and connections to other disciplines of mathematics, and likely, statistical mechanics, and combinatorial biology. We also provide a (free) translation of the paper in Polish.Comment: 27 pages, 13 figures, Paper in English with Polish translatio

Homotopy and q-homotopy skein modules of 3-manifolds: an example in Algebra Situs

Algebra Situs is a branch of mathematics which has its roots in Jones' construction of his polynomial invariant of links and Drinfeld's work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum groups, and overlaps with algebraic geometry, non-commutative geometry and statistical mechanics. Algebraic topology based on knots may be characterized as a study of properties of manifolds by considering links (submanifolds) in a manifold and their algebraic structure. The main objects of the discipline are skein modules, which are quotients of free modules over ambient isotopy classes of links in a manifold by properly chosen local (skein) relations. We concentrate, in this lecture, on one relatively simple example of a skein module of 3-manifolds -- the q-homotopy skein module. This skein module already has many ingredients of the theory: algebra structure, associated Lie algebra, quantization, state models...Comment: 40 pages. 15 Figures Dedicated to my teacher Joan Birman on her 70'th birthda

Nonorientable, incompressible surfaces in punctured-torus bundles over S^1

We classify incompressible, boundary-incompressible, nonorientable surfaces in punctured-torus bundles over $S^1$. We use the ideas of Floyd, Hatcher, and Thurston. The main tool is to put our surface in the "Morse position" with respect to the projection of the bundle into the basis S^1.Comment: 39 pages, 35 figures, one table, Dedicated to Maite Lozano for her 70th birthda

Notes to the early history of the Knot Theory in Japan

We give a description of the growth of research on Knot Theory in Japan. We place our report in a general historical context. In particular, we compare the development of research on mathematical topology in Japan with that in Poland and USA, observing several similarities. Toward the end of XIX century and at the beginning of XX century several young mathematicians, educated in Germany, France or England were returning to their native countries and building, almost from scratch, schools of modern mathematics. After a general description of the growth of topology in Japan between the World Wars, we describe the beginning of Knot Theory in Japan. Gaisi Takeuti, later a famous logician, conducted the first Knot Theory seminar in Japan in 1952 or 1953. Kunio Murasugi (later a prominent knot theoretist) was the only student who attended it. In Osaka, a Knot Theory seminar started in 1955, initiated by Hidetaka Terasaka and his students Shin'ichi Kinoshita and Takeshi Yajima. We complete the paper by listing 70 Japanese topologists born before 1946, and by sketching the biography of Fox.Comment: 29 pages. To appear in Annals of the Institute for Comparative Studies of Culture, Tokyo Woman's Christian Universit

The first coefficient of Homflypt and Kauffman polynomials: Vertigan proof of polynomial complexity using dynamic programming

We describe the polynomial time complexity algorithm for computing first coefficients of the skein (Homflypt) and Kauffman polynomial invariants of links, discovered by D.Vertigan in 1992 but never published.Comment: 8 pages, 4 figure

Fundamentals of Kauffman bracket skein modules

Skein modules are the main objects of an algebraic topology based on knots (or position). In the same spirit as Leibniz we would call our approach "algebra situs." When looking at the panorama of skein modules we see, past the rolling hills of homologies and homotopies, distant mountains - the Kauffman bracket skein module, and farther off in the distance skein modules based on other quantum invariants. We concentrate here on the basic properties of the Kauffman bracket skein module; properties fundamental in further development of the theory. In particular we consider the relative Kauffman bracket skein module, and we analyze skein modules of I-bundles over surfaces.Comment: 26 pages, 26 figure

Positive knots have negative signature

It was asked by J.Birman, Williams, and L.Rudolph whether nontrivial Lorentz knots have always positive signature. Lorentz knots are examples of positive braids (in our convention they have all crossings negative so they are negative links). It was shown by L.Rudolph that positive braids have positive signature (if they represent nontrivial links). K.Murasugi has shown that nontrivial, alternating, positive links have negative signature. Here we prove in general the old folklore conjecture that nontrivial positive links have negative signature.Comment: 5 pages, 5 figures; e-print prepared by Radmila Sazdanovi

Progress in distributive homology: from q-polynomial of rooted trees to Yang-Baxter homology

This is an extended abstract of the talk given at the Oberwolfach Workshop "Algebraic Structures in Low-Dimensional Topology", 25 May -- 31 May 2014. My goal was to describe progress in distributive homology from the previous Oberwolfach Workshop June 3 - June 9, 2012, in particular my work on Yang-Baxter homology; however I concentrated my talk on my recent discovery of q-polynomial of a rooted tree; the appropriate topic as my talk was on May 30, 2014, the 30 anniversary of the Jones polynomial, and the polynomial has its roots in the Kauffman bracket approach to the Jones polynomial.Comment: 6 pages, 8 figures; to be published in Mathematisches Forschungsinstitut Oberwolfach report 201

When the theories meet: Khovanov homology as Hochschild homology of links

We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a $(2,n)$-torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we prove the concrete connection. In the general case of Khovanov-Rozansky, $sl(n)$, homology and their deformations we conjecture the connection. The best framework to explore our ideas is to use a comultiplication-free version of Khovanov homology for graphs developed by L. Helme-Guizon and Y. Rong and extended here to to $\mathbb M$-reduced case, and to noncommutative algebras (in the case of a graph being a polygon). In this framework we prove that for any unital algebra \A the Hochschild homology of \A is isomorphic to graph homology over \A of a polygon. We expect that this paper will encourage a flow of ideas in both directions between Hochschild/cyclic homology and Khovanov homology theories.Comment: 16 pages 3 figure