Given a hyperbolic knot, we prove that the reidemeister torsion of any lift of the holonomy to sl 2. The proceeding of this conference was knots 90 by akio kawauchi, e. Reidemeisterturaev torsion of 3dimensional euler structures 15 torsion, the natural objects to deal with are euler structures with convex tangency circles. Turaev, reidemeister torsion in knot theory, uspekhi mat. The condition is the one suggested by computations for the figure eight knot example 2. The h1 ambiguity arises because the reidemeister torsion depends on a choice of ordered basis. Reidemeister torsion i reidemeister torsion is a classical topological invariant, studied since 1930s. The reidemeister torsion of 3 manifolds download ebook. This is a gaussian theory of a number of bosonic and fermionic fields in a background flat complex gl 1 gauge field. Turaev v g 1976 reidemeister torsion and the alexander polynomial mat. The present paper gives an a priori proof that this morse theory invariant is a topological invariant. The reidemeister torsion of this complex is an element of qzh1 h1, where qr denotes the total quotient ring of r. Milnor, infinite cyclic covers collected works, vol.
Reidemeister s theorem two knots are topologically equivalent if and only if their projections may be deformed into each other by a sequence of the three moves shown below. Absolute torsion and etainvariant pdf free download. Notes on the reidemeister torsion university of notre dame. The analysis of this earthquake curve can be considered as knot theory in the wide sense. This non abelian reidemeister torsion form provides a volume form on the su2representation space of g k. This paper proves a refinement of the main theorem of 1, using a different method. In topology, knot theory is the study of mathematical knots. S2, and determine the coe cient of the other component to yield a lens space.
Circlevalued morse theory and reidemeister torsion. In fact, one of the attractions of knot theory is its extensive interactions with many di erent branches of mathematics. Solodskikhholonomy observables in ponzanoreggetype state sum models john w barrett and frank hellmannrecent citations reidemeister torsion and lens surgeries on knots in homology 3spheres ii teruhisa kadokami v. The reidemeister torsion of 3manifolds request pdf. Download the reidemeister torsion of 3 manifolds ebook pdf or read online books in pdf, epub. Reidemeisterturaev torsion of 3dimensional euler structures. We prove that the twisted reidemeister torsion of a 3manifold corresponding to a fibered class is monic and we show that it gives lower bounds on the thurston norm. M hutchings, reidemeister torsion in generalized morse theory, eprint math. An international conference on knot theory and related topics was held at osaka as a satellite conference of icm kyoto in 1990. We show that appropriate assumptions on the reidemeister torsion and the cassonwalkerlescop invariant of the universal abelian covering of m imply q 1, if m is a seifert fibered space. Twisted reidemeister torsion, the thurston norm and. As an application, we obtain a new formula for turaev s combinatorial torsion. Kappeler proposed a refinement of raysinger analytic torsion which can be viewed as an analytic analogue of turaev s combinatorial torsion. Shanahan, a formula for the apolynomial of twist knots, j.
Turaev, cohomology rings, linking forms and invariants of spin structures in threedimensional manifolds, math. Chernsimons perturbation theory and reidemeister turaev torsion let p be a representation of the fundamental group of a closed oriented 3manifold m such that the corresponding local system is acyclic. In this paper, we study the topological quantum field theory that computes the reidemeister milnor turaev torsion 1, 2 in three dimensions. The twisted alexander polynomial for finite abelian covers over three. Reidemeister torsion rtorsion for short is an algebraic topology invariant which takes values in. This is a stateoftheart introduction to the work of franz reidemeister, meng taubes, turaev, and the author on the concept of torsion and its generalizations. Twisted reidemeister torsion, the thurston norm and fibered. Hyungryul baik, ursula hamenst adt and mark pedron.
The link is related to a certain subfamily of lens space surgery of. A crossing a knot is in the threedimensional space and we think that it is made of a very thin string. Reidemeister torsion was the first invariant in algebraic topology that could distinguish between closed manifolds which are homotopy equivalent but not homeomorphic, and can thus be seen as the birth of geometric topology. Lens space surgeries along certain 2component links and. We prove the mayberrymurasugi formula for links in homology 3spheres, which was proved before only for links in the 3sphere. Turaev, reidemeister torsion in knot theory, russian math. Kadokami, reidemeister torsion of seifert fibered homology lens spaces and dehn surgery, algebr. Q 1 to be the product of the zeta function and the morsetheoretic torsion. We have also avoided 4dimensional questions, such as the sliceribbon conjecture problem 1. Firstly, when the alexander polynomial of k is the same as that of a torus knot, we give a necessary and suf.
A note on reidemeister torsion and pleated surfaces. Adjoint reidemeister torsions from wrapped m5branes. In the current chapter the treatment of reidemeister torsion in 194 will be generalized to define a relative k theory invariant for chain complexes. The cobordism group of homology cylinders compositio.
During the past decade, in the work of vladimir turaev, new points of view have emerged, which turned out to be the right ones as far as gauge theory. For hyperbolic twist knots, we also prove that the nonabelian reidemeister torsion at the holonomy representation can be expressed as a rational function evaluated at the cusp shape of the knot. The present paper gives an a priori proof that this morse theory invariant is a topological in. In 1992, the journal of knot theory and its ramifications was founded. Refined analytic torsion of lens spaces semantic scholar. In the three dimensional case, combining this result with the conjecture in 1, we obtain a formula for the full seibergwitten invariant, which was conjectured by turaev. Johnson, a geometric form of casson invariant and its connection to reidemeister torsion, unpublished. This theory and its connection to seibergwitten invariants. Seifert surgery on knots via reidemeister torsion and cassonwalkerlescop invariant ii kadokami, teruhisa, maruyama, noriko, and sakai, tsuyoshi, osaka journal of mathematics, 2016. This is simply the torsion of a simplicial complex with abelian local coef. Reidemeister torsion the alexander polynomial and u1 1.
Kadokami, reidemeister torsion and lens surgeries on knots in homology 3spheres i, osaka j. The first is an understanding of how the components of the invariant change when we remove a curve from the singular set. It is hoped that this will provide a model for possible generalizations to floer theory. Reidemeister torsion for the morse theory of closed 1forms on a. They correspond to the simplest changes in a diagram that occur when a knot is. The reidemeister torsion of an arbitrary simplicial or cw complex is defined in. Turaev torsion invariants of 3orbifolds biji wong 2016 geometriae dedicata. A deformation of the alexander polynomials of knots. Reidemeister torsion and dehn surgery on twist knots. Torsion is the oldest topological but not with respect to homotopy invariant that in its almost eight decades of existence has been at the center of many important and surprising discoveries.
Retrieve articles in proceedings of the american mathematical society with msc 2010. The knot group of a knot awith base point b2s3 ima is the fundamental group of the knot complement of a, with bas the base point. In the case of link complements, this torsion can be used. Sep 26, 20 twisted reidemeister torsion, the thurston norm and fibered manifolds twisted reidemeister torsion, the thurston norm and fibered manifolds friedl, stefan 20926 00. The torsion of a cellular simplicial complex was introduced in the 30s by w. There are many connections between reidemeister torsion and knots, particularly the alexander polynomials of knots see milnor 193, 194, 195 and turaev 293, as well as chap. The torsion in question was intro duced by milnor 5. Yamaguchi, nonabelian reidemeister torsion for twist knots, j. Reidemeister torsion and lens surgeries on knots in homology. Singer 1971, 1973a, 1973b as an analytic analogue of reidemeister torsion. Turaev, reidemeister torsion in knot theory russian math. In the current chapter the treatment of reidemeister torsion in 194 will be generalized to define a relative ktheory invariant for chain complexes. Chapter 3 focuses on turaevs ingenious idea of euler structure and how it can.
It is not usually at all obvious how to determine whether one knot an embedding in three dimensions of a. As the name suggests it is an introductory book in graduate level about knots. Download pdf the reidemeister torsion of 3 manifolds. Refined torsion and the refined alexander function 146 4. Eudml non abelian reidemeister torsion and volume form. An introduction to knot theory and the knot group 5 complement itself could be considered a knot invariant, albeit a very useless one on its own. A reidemeister move is an operation that can be performed on the diagram of a knot whithout altering the corresponding knot.
This cited by count includes citations to the following articles in scholar. Turaev 27 showed that the h1 ambiguity can be resolved by the choice of an euler structure. I twisted reidemeister torsion associated with a representation of the fundamental group to gln,f has been studied since the early 1990s. Some applications, including relations between the reidemeister torsion and other classical invariants, are surveyed. Russian articles, english articles this publication is cited in the following articles. Stefan friedl 2016 journal of knot theory and its ramifications 17400. The second is a formula relating the invariant of the 3orbifold to the turaev torsion invariant of the underlying 3manifold in the case when the singular set is a nullhomologous knot. Foxs example 12 from a quick trick in knot theory is the 2twist spun trefoil. K of knot theory by applying the above either to the knot complement or to the 3manifold obtained by doing 0surgery along k. Seifert surgery on knots via reidemeister torsion and casson. Consider a knot j, and a second knot k created by going around the knot j n times, following the shape of j then k is the satellite knot with companion j the winding number of k is n theorem li, guo.
Nonabelian reidemeister torsion for twist knots journal of. Using signrefined torsion, we derive a skein relation for a normalized form of this polynomial. Full pdf abstract top these notes accompany some lectures given at the autumn school tresses in pau in october 2009. Although these do have a signi cant in uence on elementary knot theory, via unknotting number. We also compute the refined analytic torsion of lens. Jeff cheeger 1977, 1979 and werner muller 1978 proved ray and singers conjecture that reidemeister torsion and analytic torsion are the. Relations between torsion and the alexander invariant are well known. Kurt reidemeister and, independently, james waddell alexander and garland baird briggs, demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a sequence of the three reidemeister moves. It is a fortunate fact, peculiar of dimension 3, that there is a canonical way to associate a convex. Farber department of mathematics, tel aviv university, tel aviv, 69978, israel email. An introduction to the abelian reidemeister torsion of. We plan a reading seminar to learn about reidemeister torsion and its applications, with a view to understanding some of the following papers.
Notes on reidemeister torsion reidemeister torsion r. Reidemeister torsion and lens surgeries on knots in. Turaev, reidemeister torsion and the alexander polynomial. We give an invariant dp of p as a 1dimensional cohomology class of m with twisted coefficient.
In the mathematical area of knot theory, a reidemeister move is any of three local moves on a link diagram. So the alexander polynomial, as a single polynomial, is not defined. The present volume grew out of the heidelberg knot theory semester, organized by. Twisted alexander polynomials, character varieties and reidemeister torsions of double branched covers yoshikazu yamaguchi 2016 topology and its applications 204 278.
Reidemeister torsion in generalized morse theory math berkeley. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined together so that it cannot be undone, the simplest knot being a ring or unknot. This is a gaussian theory of a number of bosonic and. A closely related invariant is the reidemeistermilnorturaev torsion m. Reidemeister torsion and group invariants of threedimensional manifolds. We use reidemeister torsion to study a twisted alexander polynomial, as defined by turaev, for links in the projective space. The former fixes a flawed proof in friedl and vidussi 2010, the latter gives a quick alternative argument for the main theorem of friedl and kim topology 45. Abstract references similar articles additional information. Here we are interested in the adjoints of representations to sl. Circlevalued morse theory and reidemeister torsion arxiv.
Analytic torsion or raysinger torsion is an invariant of riemannian manifolds defined by daniel b. Turaev 6, we defined a new concept of combinatorial torsion which we called absolute torsion. S3 is a knot we obtain the usual alexander polynomial. Refined torsion and the refined alexander function 4. In this paper, we study the topological quantum field theory that computes the reidemeister milnorturaev torsion 1, 2 in three dimensions. Seifert surgery on knots via reidemeister torsion and. The ideals that define the homology of the infinite cyclic covering are not principal. Reidemeister torsion and group invariants of three. Available formats pdf please select a format to send.