arXiv subjects
Publications and source records attributed to Jozef H. Przytycki.
We show how the periodicity of a homology sphere is reflected in the Reshetikhin-Turaev-Witten invariants of the manifold. These yield a criterion for the periodicity of a homology sphere.
We study the linking numbers in a rational homology 3-sphere and in the infinite cyclic cover of the complement of a knot. They take values in $\Bbb Q$ and in ${Q}({\Bbb Z}[t,t^{-1}])$ respectively, where ${Q}({\Bbb Z}[t,t^{-1}])$ denotes the quotient field of ${\Bbb Z}[t,t^{-1}]$. It is known that the modulo-$\Bbb Z$ linking number in the rational homology 3-sphere is determined by the linking matrix of the framed link and that the modulo-${\Bbb Z}[t,t^{-1}]$ linking number in the infinite cyclic cover of the complement of a knot is determined by the Seifert matrix of the knot. We eliminate ` modulo $\Bbb Z$' and ` modulo ${\Bbb Z}[t,t^{-1}]$'. When the finite cyclic cover of the 3-sphere branched over a knot is a rational homology 3-sphere, the linking number of a pair in the preimage of a link in the 3-sphere is determined by the Goeritz/Seifert matrix of the knot.
L. Kauffman conjectured that a particular solution of the Chinese Rings puzzle is the simplest possible. We prove his conjecture by using low-dimensional topology and group theory. We notice also a surprising connection between the Chinese Rings and Habiro moves (related to Vassiliev invariants).
Talk 1: Open problems in knot theory that everyone can try to solve. Knot theory is more than two hundred years old; the first scientists who considered knots as mathematical objects were A.Vandermonde (1771) and C.F.Gauss (1794). However, despite the impressive grow of the theory, there are simply formulated but fundamental questions, to which we do not know answers. I will discuss today several such open problems, describing in detail the 20 year old Montesinos-Nakanishi conjecture. Our problems lead to sophisticated mathematical structures (I will describe some of them in tomorrows talks), but today's description will be absolutely elementary. Talk 2: Lagrangian approximation of Fox $p$-colorings of tangles. The talk will culminate in the introduction of the symplectic structure on the boundary of a tangle in such a way that tangles yields Lagrangians in the symplectic space. Talk 3: Historical Introduction to Skein Modules. I will discuss, in my last talk of the conference, skein modules, or as I prefer to say more generally, algebraic topology based on knots. I would like to discuss today, in more detail, skein modules related to the (deformations) of 3-moves and the Montesinos-Nakanishi conjecture but first I will give the general definition and I will make a short tour of the world of skein modules.
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.
We give a concise proof of a classification of lens spaces up to orientation-preserving homeomorphisms. The chief ingredient in our proof is a study of the Alexander polynomial of ` symmetric' links in $S^3$.
We study the concept of the fourth skein module of 3-manifolds, that is a skein module based on the skein relation b_0L_0 + b_1L_1 + b_2L_2 + b_3L_3 = 0 and a framing relation L^{(1)} = aL, where a, b_0, b_3 are invertible. We introdule the concept of n-algebraic tangles (and links) and analyze the skein module for 3-algebraic links. As a byproduct we prove the Montesinos-Nakanishi 3-move conjecture for 3-algebraic links (including 3-bridge links).
We show that a closed orientable 3-manifold M admits an action of Z_p with fixed point set S^1 iff M can be obtained as the result of surgery on a p-periodic framed link L and Z_p acts freely on the components of L. We prove a similar theorem for free Z_p-actions. As an interesting application, we prove the following, rather unexpected result: for any M as above and for any odd prime p, H_1(M, Z_p)\ne Z_p. We also prove a similar criterion of 2-periodicity for rational homology 3-spheres.
We show that for the Kauffman bracket skein module over the field of rational functions in variable A, the module of a connected sum of 3-manifolds is the tensor product of modules of the individual manifolds.
We describe, for a few small examples, the Kauffman bracket skein algebra of a surface crossed with an interval. If the surface is a punctured torus the result is a quantization of the symmetric algebra in three variables (and an algebra closely related to a cyclic quantization of $U(so_3$). For a torus without boundary we obtain a quantization of "the symmetric homologies" of a torus (equivalently, the coordinate ring of the $SL_2(C)$-character variety of $Z \oplus Z$). Presentations are also given for the four punctured sphere and twice punctured torus. We conclude with an investigation of central elements and zero divisors.
We give a short proof for a formula for the number of divisions of a convex (sn+2)-gon along non-crossing diagonals into (sj+2)-gons, where 1<=j<=n-1. In other words, we consider dissections of an (sn+2)-gon into pieces which can be further subdivided into (s+2)-gons. This formula generalizes the formulas for classical numbers of polygon dissections: Euler-Catalan number, Fuss number and Kirkman-Cayley number. Our proof is elementary and does not use the method of generating functions.
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.
This paper gives insight into intriguing connections between two apparently unrelated theories: the theory of skein modules of 3-manifolds and the theory of representations of groups into special linear groups of 2 by 2 matrices. Let R be a ring with an invertible element A. For any 3-manifold M one can assign an R-module called the Kauffman bracket skein module of M. If A^2=1 then this module has a structure of an R-algebra. We investigate this structure and, in particular, we prove that if R is the field of complex numbers then this algebra is isomorphic to the (unreduced) coordinate ring of the SL_2-character variety of pi_1(M). Using that result we develop a theory of Sl_2-character varieties by use of topological methods. We also assign to any surface a relative Kauffman bracket skein algebra. We prove several results about this non-commutative algebra. Our work should be considered in the context of the book of Brumfiel and Hilden `SL(2) Representations of Finitely Presented Groups,' Cont. Math 187. In particular we give a topological interpretation to algebraic objects considered in that book.