Jozef H. Przytycki

THE (2, ∞)-SKEIN MODULE OF LENS SPACES: A GENERALIZATION OF THE JONES POLYNOMIAL

We extend the Jones polynomial for links in S3 to links in L(p, q), p>0. Specifically, we show that the (2, ∞)-skein module of L(p, q) is free with [p/2]+1 generators. In the case of S1×S2 the skein module is infinitely generated.

Conway algebras and skein equivalence of links

We consider a class of pairs of links which are not skein equivalent but have the same invariant in every Conway algebra. 1. Conway algebras. We will first recall the notion of Conway algebra as introduced in [PT]. DEFINITION 1.1. A Conway algebra is an algebra A with a sequence of 0argument operations a1, a2, ... and two 2-argument operations I and *, which satisfy the following conditions: Ci. anIan+1 = ani C2. an * an+1 = an (Cl and C2 are initial conditions properties), C3. (alb)I(cld) = (alc)I(bld), C4. (aIb) * (cId) = (a * c)I(b * d) (C3, C4 and C5 are transposition properties), C5. (a*b) * (c*d) = (a*c) * (b*d), C6. (alb) * b = a, C7. (a*b)lb=a. As shown in [PT] every Conway algebra yields an invariant of links which is constant on skein equivalence classes (skein invariant). It is uniquely determined by the following conditions: AT, = an (initial relations), AL+ = AL_ JALo and AL_ = AL+ * ALO (Conway relations). Here Tn denotes a trivial link of n components and L+, L-, and LO are diagrams of oriented links identical except near one crossing point (see Figure 1.1).

3-coloring and other elementary invariants of knots

Classical knot theory studies the position of a circle (knot) or of several circles (link) in R or S = R3∪∞. The fundamental problem of classical knot theory is the classification of links (including knots) up to the natural movement in space which is called an ambient isotopy. To distinguish knots or links we look for invariants of links, that is, properties of links which are unchanged under ambient isotopy. When we look for invariants of links we have to take into account the following three criteria:

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.

Knot polynomials and generalized mutation

Abstract 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 Conways 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.

Distributivity versus associativity in the homology theory of algebraic structures

Abstract While homology theory of associative structures, such as groups and rings, has been extensively studied in the past beginning with the work of Hopf, Eilenberg, and Hochschild, homology of non-associative distributive structures, such as quandles, were neglected until recently. Distributive structures have been studied for a long time. In 1880, C.S. Peirce emphasized the importance of (right) self-distributivity in algebraic structures. However, homology for these universal algebras was introduced only sixteen years ago by Fenn, Rourke, and Sanderson. We develop this theory in the historical context and propose a general framework to study homology of distributive structures. We illustrate the theory by computing some examples of 1-term and 2-term homology, and then discussing 4-term homology for Boolean algebras. We outline potential relations to Khovanov homology, via the Yang-Baxter operator.

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 coverings of S3 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 S3 branched along various hyperbolic 2-bridge knots. The manifold obtained in such a way from the 52 knot is of special interest as it is conjectured to be the hyperbolic 3-manifold with the smallest volume.

Lissajous knots and billiard knots

We show that Lissajous knots are equivalent to billiard knots in a cube. We consider also knots in general 3-dimensional billiard tables. We analyse symmetry of knots in billiard tables and show in particular that the Alexander polynomial of a Lissajous knot is a square modulo 2. 0. Introduction. A Lissajous knot K is a knot in R given by the parametric equations x = cos(ηxt+ φx), y = cos(ηyt+ φy), z = cos(ηzt+ φz), for integers ηx, ηy, ηz. A Lissajous link is a collection of disjoint Lissajous knots. The fundamental question was asked in [BHJS94]: which knots are Lissajous? One defines a billiard knot (or racquetball knot) as the trajectory inside a cube of a ball which leaves a wall at rational angles with respect to the natural frame, and travels in a straight line except for reflecting perfectly off the walls; generically it will miss the corners and edges, and will form a knot. We will show that these knots are precisely the same as the Lissajous knots. We will also speculate about more general billiard knots, e.g. taking another polyhedron instead of the ball, considering a non-Euclidean metric, or considering the trajectory of a ball in the configuration space of a flat billiard. We will illustrate these by various examples. For instance, the trefoil knot is not a Lissajous knot 1991 Mathematics Subject Classification: 57M25, 58F17. This is an extended version of the talk given in August 1995, at the minisemester on Knot Theory at the Banach Center. We would like to acknowledge the support from USAF grant 1-443964-22502. The paper is in final form and no version of it will be published elsewhere.

The skein polynomial of a planar star product of two links

If P L (v,z) = Σ b i (v)z i is the skein polynomial of a link L , and D = D 1 * D 2 is the diagram which is a planar star (Murasugi) product of D 1 and D 2 then b ϕ(D) (v) = b ϕ(D 1 ) · b ϕ(D 2 ) (v) where ϕ(D) = n(D)– (s(D) – 1) and n(D) denotes the number of crossings of D , and s(D) is the number of Seifert circles of D .

Incompressible surfaces in the exterior of a closed 3-braid

Let be a closed 3-braid in S 3 with axis L γ . The exterior of γ U Lγ in S 3 is a fibre bundle over S 1 with fibre a disc with 3 holes. The monodromy is given by a matrix in PSL(2, ) and the braid γ is called hyperbolic if its matrix is hyperbolic.