Searcharxiv⌕ Search

arXiv subjects

Jozef H. Przytycki

Publications and source records attributed to Jozef H. Przytycki.

At least 73 records · Page 4Linked to original sources
arXiv2005

Burnside kei

This paper is motivated by a general question: for which values of k and n is the universal Burnside kei of k generators and Kei "exponent" n, $\bar Q(k,n)$, finite? It is known (starting from the work of M. Takasaki (1942)) that $\bar Q(2,n)$ is isomorphic to the dihedral quandle Z_n and $\bar Q(3,3)$ is isomorphic to Z_3 + Z_3. In this paper we give descriptions of $\bar Q(4,3)$ and $\bar Q(3,4)$. We also investigate some properties of arbitrary quandles satisfying the universal Burnside relation (of "exponent" n) a=...a*b*...*a*b. In particular, we prove that the order of a finite commutative kei is a power of 3. Invariants of links related to Burnside kei $\bar Q(k,n)$ are invariant under n-moves.

math.GT↗
arXiv2005

Khovanov Homology: categorification of the Kauffman bracket relation

Khovanov homology offers a nontrivial generalization of Jones polynomial of links in R^3 (and of Kauffman bracket skein module of some 3-manifolds). In this chapter (Chapter X) we define Khovanov homology of links in R^3 and generalize the construction into links in an I-bundle over a surface. We use Viro's approach to construction of Khovanov homology and utilize the fact that one works with unoriented diagrams (unoriented framed links) in which case there is a simple description of Khovanov long exact sequence of homology. Khovanov homology, over the field Q, is a categorification of the Jones polynomial (i.e. one represents the Jones polynomial as the generating function of Euler characteristics). However, for integral coefficients Khovanov homology almost always has torsion. The first part of the chapter is devoted to the construction of torsion in Khovanov homology. In the second part we analyze the thickness of Khovanov homology and reduced Khovanov homology. In the last part we generalize Khovanov homology to links in $I$-bundles over surfaces and demonstrate that for oriented surfaces we categorify the element of the Kauffman bracket skein module represented by the considered link.

math.GT↗
arXiv2005

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.

math.GT↗
arXiv2005

Rational moves and tangle embeddings: (2,2)-moves as a case study

We classify 3-braids up to (2,2)-move equivalence and in particular we show how to adjust the Harikae-Nakanishi-Uchida conjecture so it holds for closed 3-braids. As important steps to classify 3-braids up to (2,2)-move equivalence we prove the conjecture for 2-algebraic links and classify (2,2)-equivalence classes for links up to nine crossings. We analyze the behavior of Kei (involutive quandle) associated to a link under (2,2)-moves. We construct Burnside Kei, Q(m,n), and ask the question, motivated by classical Burnside question: for which values of m and n, is Q(m,n) finite?

math.GT↗
arXiv2004

Categorification of the Kauffman bracket skein module of I-bundles over surfaces

Khovanov defined graded homology groups for links L in R^3 and showed that their polynomial Euler characteristic is the Jones polynomial of L. Khovanov's construction does not extend in a straightforward way to links in I-bundles M over surfaces F not D^2 (except for the homology with Z/2 coefficients only). Hence, the goal of this paper is to provide a nontrivial generalization of his method leading to homology invariants of links in M with arbitrary rings of coefficients. After proving the invariance of our homology groups under Reidemeister moves, we show that the polynomial Euler characteristics of our homology groups of L determine the coefficients of L in the standard basis of the skein module of M. Therefore, our homology groups provide a `categorification' of the Kauffman bracket skein module of M. Additionally, we prove a generalization of Viro's exact sequence for our homology groups. Finally, we show a duality theorem relating cohomology groups of any link L to the homology groups of the mirror image of L.

math.QA↗
arXiv2004

Signature of rotors

Rotors were introduced in Graph Theory by W.Tutte. The concept was adapted to Knot Theory as a generalization of mutation by Anstee, Przytycki and Rolfsen in 1987. In this paper we show that Tristram-Levine signature is preserved by orientation-preserving rotations. Moreover, we show that any link invariant obtained from the characteristic polynomial of Goeritz matrix, including Murasugi signature, is not changed by rotations. In 2001, P. Traczyk showed that the Conway polynomials of any pair of orientation-preserving rotants coincide. But it was still an open problem if an orientation-reversing rotation preserves Conway polynomial. We show that there is a pair of orientation-reversing rotants with different Conway polynomials. This provides a negative solution to the problem.

math.GT↗
arXiv2004

Khovanov homology: torsion and thickness

We partially solve the conjecture by A.Shumakovitch about torsion in the Khovanov homology of prime, non-split links in S^3. We give a size restriction on the Khovanov homology of almost alternating links. We relate the Khovanov homology of the connected sum of a link diagram and the Hopf link with the Khovanov homology of the diagram via a short exact sequence of homology which splits. Finally we show that our results can be adapted to reduced Khovanov homology and we show that there is a long exact sequence connecting reduced Khovanov homology with unreduced homology.

math.GT↗
arXiv2004

Search for different links with the same Jones' type polynomials: Ideas from graph theory and statistical mechanics

We describe in this talk three methods of constructing different links with the same Jones type invariant. All three can be thought as generalizations of mutation. The first combines the satellite construction with mutation. The second uses the notion of rotant, taken from the graph theory, the third, invented by Jones, transplants into knot theory the idea of the Yang-Baxter equation with the spectral parameter (idea employed by Baxter in the theory of solvable models in statistical mechanics). We extend the Jones result and relate it to Traczyk's work on rotors of links. We also show further applications of the Jones idea, e.g. to 3-string links in the solid torus. We stress the fact that ideas coming from various areas of mathematics (and theoretical physics) has been fruitfully used in knot theory, and vice versa. (This is the detailed version of the talk given at the Banach Center Colloquium, Warsaw, Poland, March 24, 1994: ``W poszukiwaniu nietrywialnego wezla z trywialnym wielomianem Jonesa: grafy i mechanika statystyczna")

math.GT↗
arXiv2004

3-Manifolds, tangles and persistent invariants

New obstructions for embedding one compact oriented 3-manifold in another are given. A theorem of D. Krebes concerning 4-tangles embedded in links arises as a special case. Algebraic and skein-theoretic generalizations for 2n-tangles provide invariants that persist in the corresponding invariants of links in which they embed. An example is given of a virtual 4-tangle for which Krebes's theorem does not hold.

math.GT↗
arXiv2004

Knot polynomials and generalized mutation

The motivation for this work was to construct a nontrivial knot with trivial Jones polynomial. Although that open problem has not yielded, the methods are useful for other problems in the theory of knot polynomials. The subject of the present paper is a generalization of Conway's mutation of knots and links. Instead of flipping a 2-strand tangle, one flips a many-string tangle to produce a generalized mutant. In the presence of rotational symmetry in that tangle, the result is called a "rotant". We show that if a rotant is sufficiently simple, then its Jones polynomial agrees with that of the original link. As an application, this provides a method of generating many examples of links with the same Jones polynomial, but different Alexander polynomials. Various other knot polynomials, as well as signature, are also invariant under such moves, if one imposes more stringent conditions upon the symmetries. Applications are also given to polynomials of satellites and symmetric knots.

math.GT↗
arXiv2004

Non-left-orderable 3-manifold groups

We show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched covers of S^3 branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in these examples are known as Fibonacci groups which we show not to be left-orderable. Many other examples of non-orderable groups are obtained by taking 3-fold branched covers of S^3 branched along various hyperbolic 2-bridge knots. The manifold obtained in such a way from the 5_2 knot is of special interest as it is conjectured to be the hyperbolic 3-manifold with the smallest volume.

math.GT↗
arXiv2004

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.

math.GT↗
arXiv2004

The topological interpretation of the core group of a surface in S^4

We give a topological interpretation of the core group invariant of a surface embedded in S^4. We show that the group is isomorphic to the free product of the fundamental group of the double branch cover of S^4 with the surface as a branched set, and the infinite cyclic group. We present a generalization for unoriented surfaces, for other cyclic branched covers, and other codimension two embeddings of manifolds in spheres. The method of computing the fundamental group of n-fold cyclic branched covers is related to the one described in R.H.Crowell, The derived group of a permutation representation, Adv. in Math. 53(1), 1984, 99--124. We use these computations in recent papers: http://front.math.ucdavis.edu/math.GT/0302098 http://front.math.ucdavis.edu/math.GT/0309140

math.GT↗
arXiv2004

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...

math.GT↗
arXiv2003

Unexpected connections between Burnside Groups and Knot Theory

In Classical Knot Theory and in the new Theory of Quantum Invariants substantial effort was directed toward the search for unknotting moves on links. We solve, in this note, several classical problems concerning unknotting moves. Our approach uses a new concept, Burnside groups of links, which establishes unexpected relationship between Knot Theory and Group Theory. Our method has the potential to be used in computational biology in the analysis of DNA via tangle embedding theory, as developed by D.W.Sumners.

math.GT↗
arXiv2003

Kauffman-Harary conjecture holds for Montesinos Knots

The Kauffman-Harary conjecture states that for any reduced alternating diagram K of a knot with a prime determinant p, every non-trivial Fox p-coloring of K assigns different colors to its arcs. We generalize the conjecture by stating it in terms of homology of the double cover of S^3 branched along a link. In this way we extend the scope of the conjecture to all prime alternating links of arbitrary determinants. We first prove the Kauffman-Harary conjecture for pretzel knots and then we generalize our argument to show the generalized Kauffman-Harary conjecture for all Montesinos links. Finally, we speculate on the relation between the conjecture and Menasco's work on incompressible surfaces in exteriors of alternating links.

math.GT↗