Turaev v g 1976 reidemeister torsion and the alexander polynomial mat. The analysis of this earthquake curve can be considered as knot theory in the wide sense. 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. But sometimes it is so confusing that i dont know which type move should i perform on link to get desired result. Because of reidemeisters theorem, theorem, they are of fundamental importance in knot theory, allowing knots and links to be treated purely combinatorially and diagrammatically. Reidemeister torsion is a classical yet very vital topic in 3dimensional topology, and it was recently used in a variety of important developments. Nonabelian reidemeister torsion for twist knots request pdf. In this paper, we study the topological quantum field theory that computes the reidemeister milnorturaev torsion 1, 2 in three dimensions. Firstly, when the alexander polynomial of k is the same as that of a torus knot, we give a necessary and suf.
The reidemeister moves are local changes to link diagrams which can be realised as isotopies between the link diagrams from which they arise. Here we are interested in the adjoints of representations to sl. This is a gaussian theory of a number of bosonic and fermionic fields in a background flat complex gl 1 gauge field. In the current chapter the treatment of reidemeister torsion in 194 will be generalized to define a relative k theory invariant for chain complexes.
Our mentor, zhenkun li, for teaching us about knot theory. He coedited the journal mathematische annalen from 1947 until 1963. This is simply the torsion of a simplicial complex with abelian local coef. Seifert surgery on knots via reidemeister torsion and. The reidemeister torsion of this complex is an element of qzh1 h1, where qr denotes the total quotient ring of r. You know about manipulating knots and how that can be reflected in the group presentations of the knot group. To mention a few, torsion is a fundamental ingredient of the cassonwalkerlescop invariants see e. The discovery of the jones polynomial by vaughan jones in 1984 not only showed a connection between knot theory and di erent areas of mathematics operator algebras, braid theory, quantum groups, but also to physics statistical models 2, 8.
The proceeding of this conference was knots 90 by akio kawauchi, e. A relationship between the nonacyclic reidemeister torsion. Reidemeister torsion has a very useful application in knot theory and links. The h1 ambiguity arises because the reidemeister torsion depends on a choice of ordered basis. Kadokami, reidemeister torsion and lens surgeries on knots in homology 3spheres i, osaka j. Reidemeister torsion in knot theory 121 these connections, to give a survey of the main properties of alexander polynomials of links in the threedimensional sphere. We show a relationship between the nonacyclic reidemeister torsion and a zero of the acyclic reidemeister torsion for a. Reidemeister torsion and lens surgeries on knots in homology 3spheres ii teruhisa kadokami 2008 topology and its applications 155 1699. Twisted alexander polynomials, character varieties and reidemeister torsions of double branched covers yoshikazu yamaguchi 2016 topology and its applications 204 278. Pdf reidemeister torsion in knot theory researchgate. It is not usually at all obvious how to determine whether one knot an embedding in three dimensions of a closed loop is equivalent to another. Milnor j w 1962 a duality theorem for reidemeister torsion ann. Reidemeisters 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.
Stefan friedl 2016 journal of knot theory and its ramifications 17400. Reidemeister torsion and volume for hyperbolic knots p. Reidemeister torsion for the morse theory of closed 1forms on a. Jaewook chung and xiaosong lin 2009 journal of knot theory and its ramifications jktr 18 1227. Although these do have a signi cant in uence on elementary knot theory, via unknotting number. Is there a geometric interpretation for reidemeister torsion. A relationship between the nonacyclic reidemeister torsion and a zero of the acyclic reidemeister torsion tome 58, no 1 2008, p. Request pdf twisted reidemeister torsion for twist knots the aim of this paper is to give an explicit formula for the sl2ctwisted reidemeister torsion as defined in 6 in the case of twist. An international conference on knot theory and related topics was held at osaka as a satellite conference of icm kyoto in 1990. It is hoped that this will provide a model for possible generalizations to floer theory.
I twisted reidemeister torsion associated with a representation of the fundamental group to gln,f has been studied since the early 1990s. Reidemeister torsion i reidemeister torsion is a classical topological invariant, studied since 1930s. A crossing a knot is in the threedimensional space and we think that it is made of a very thin string. A relationship between the nonacyclic reidemeister.
Analytic torsion or raysinger torsion is an invariant of riemannian manifolds defined by daniel b. If one knot diagram can be turned into another using any sequence of the reidemeister moves, in combination with planar isotopies deformation then they belong to the same knot class. It gives an explicit formula of the reidemeister torsion of knot exteriors. In the mathematical area of knot theory, a reidemeister move is any of three local moves on a link diagram. Turaev 27 showed that the h1 ambiguity can be resolved by the choice of an euler structure. Although highdimensional knot theory does not have such glamorous applications as classical knot theory, it has many fascinating results of its own, which make use of a wide variety of sophisticated algebraic and geometric methods. Kadokami, reidemeister torsion of seifert fibered homology lens spaces and dehn surgery, algebr. Reidemeisters interests were mainly in combinatorial group theory, combinatorial topology, geometric group theory, and the foundations of geometry.
It is not usually at all obvious how to determine whether one knot an embedding in three dimensions of a. This approach makes it possible to represent the numerous and diverse properties of alexander polynomials of links as consequences of several universal torsion properties. Reidemeister torsion and lens surgeries on knots in homology. Relations have been pointed out between torsion and hyperbolic geometry. They correspond to the simplest changes in a diagram that occur when a knot is. By 1970, knot theory had become a welldeveloped area of topology 7. Reidemeister developed both the geometric and some of the algebraic side extending that sort of combinatorial theory.
In 1992, the journal of knot theory and its ramifications was founded. Reidemeister torsion the alexander polynomial and u1. Reidemeister torsion and lens surgeries on knots in. Nonabelian reidemeister torsion for twist knots journal of. Refined torsion and the refined alexander function 4. Pdf a note on reidemeister torsion and pleated surfaces. Saleury department of physics university of texas at austin austin, tx 78712 ydepartment of physics yale university new haven, ct 06519 abstract. 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. Reidemeister torsion of the figureeight knot exterior for. A reidemeister move is an operation that can be performed on the diagram of a knot whithout altering the corresponding knot. Jeff cheeger 1977, 1979 and werner muller 1978 proved ray and singers conjecture that reidemeister torsion and analytic torsion are the.
The present paper gives an a priori proof that this morse theory invariant is a topological invariant. 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. Turaev torsion invariants of 3orbifolds biji wong 2016 geometriae dedicata. Reidemeister torsion and volume for hyperbolic knots. Twisted reidemeister torsion for twist knots request pdf. Supposec m pl c m1 pl 2pl c 0 is an acycliccomplexof. It has also applications in dynamical systems see, e. Seifert surgery on knots via reidemeister torsion and cassonwalkerlescop invariant ii kadokami, teruhisa, maruyama, noriko, and sakai, tsuyoshi, osaka journal of mathematics, 2016.
Seifert surgery on knots via reidemeister torsion and casson. 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 mathematical language, a knot is an embedding of a circle in 3dimensional euclidean space, r 3 in. There is sense in which it all comes down to a generalisation of rewriting theory. Choose u i 3c i for each i so that 0olu i1 u i 3501c i.
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. In topology, knot theory is the study of mathematical knots. Singer 1971, 1973a, 1973b as an analytic analogue of reidemeister torsion. Pdf reidemeister torsion, twisted alexander polynomial. Reidemeisterturaev torsion of 3dimensional euler structures. Visualizing and exploring knot theory with fluid reidemeister moves hui zhang, jianguang weng, lin jing, and yiwen zhong a b fig.
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