Conformal invariance of the writhe of a knot (original) (raw)
Conformally invariant energies of knots II
2004
This paper has been withdrawn by the authors, as it was combined with "Conformally invariant energies of knots I" (math/0409396) to be "Conformally invariant energies of knots" which has replaced the former.
Conformally invariant energies of knots
2004
Conformally invariant functionals on the space of knots are introduced via extrinsic conformal geometry of the knot and integral geometry on the space of spheres. Our functionals are expressed in terms of a complex-valued 2-form which can be considered as the cross-ratio of a pair of infinitesimal segments of the knot. We show that our functionals detect the unknot as the total curvature does, and that their values explode if a knot degenerates to a singular knot with double points.
DUALITY PROPERTIES OF INDICATRICES OF KNOTS
The bridge index and superbridge index of a knot are important invariants in knot theory. We define the bridge map of a knot conformation, which is closely related to these two invariants, and interpret it in terms of the tangent indicatrix of the knot conformation. Using the concepts of dual and derivative curves of spherical curves as introduced by Arnold, we show that the graph of the bridge map is the union of the binormal indicatrix, its antipodal curve, and some number of great circles. Similarly, we define the inflection map of a knot conformation, interpret it in terms of the binormal indicatrix, and express its graph in terms of the tangent indicatrix. This duality relationship is also studied for another dual pair of curves, the normal and Darboux indicatrices of a knot conformation. The analogous concepts are defined and results are derived for stick knots.
2006
We study a 1-form which can be given by a vector in a conformally invariant way. We then study conformally invariant functionals associated to a ``Y-diagram'' on the space of knots which are made from the 1-form.
A one-parameter approach to knot theory
2006
To an oriented knot we associate a trace graph in a thickened torus in such a way that knots are isotopic if and only if their trace graphs can be related by moves of finitely many standard types. For closed braids with a fixed number of strands, we recognize trace graphs up to equivalence excluding one type of moves in polynomial time with respect to the braid length.
An unknotting invariant for welded knots
Proceedings - Mathematical Sciences, 2021
We study a local twist move on welded knots that is an analog of the virtualization move on virtual knots. Since this move is an unknotting operation we define an invariant, unknotting twist number, for welded knots. We relate the unknotting twist number with warping degree and welded unknotting number, and establish a lower bound on the twist number using Alexander quandle coloring. We also study the Gordian distance between welded knots by twist move and define the corresponding Gordian complex. 2010 Mathematics Subject Classifications. 57M25, 57M27.
An Invariant for Singular Knots
Journal of Knot Theory and Its Ramifications, 2009
In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma–Hecke algebras Y d,n(u) and the theory of singular braids. The Yokonuma–Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid SBn into the algebra Y d,n(u). Surprisingly, the trace does not normalize directly to yield a singular link invariant, so a condition must be imposed on the trace variables. Assuming this condition, the invariant satisfies a skein relation involving singular crossings, which arises from a quadratic relation in the algebra Y d,n(u).
The L^2 signature of torus knots
2010
We find a formula for the L2 signature of a (p,q) torus knot, which is the integral of the omega-signatures over the unit circle. We then apply this to a theorem of Cochran-Orr-Teichner to prove that the n-twisted doubles of the unknot, for n not 0 or 2, are not slice. This is a new proof of the result first proved by Casson and Gordon.
Journal of Differential Geometry, 1995
Let 2 c C n+ι be an algebraic (analytic) hypersurface with an isolated singularity at the origin, which is given as the zero set of /: C n+ι-> C. Recall that the link of such a singularity (5 2/I+1 , K 2n~ι) consists of a highly connected manifold K, embedded in the sphere S, as a codimensiontwo submanifold. Moreover, the complement S-K of this embedding fibers over the circle, with the projection map given by the Milnor fibration /(z)/|/(z). Thus these knots belong to a larger class of knots known as simple fibered knots. From one point of view, simple fibered knots are more general than the objects of study in spherical knot theory, since the submanifold K need not be a sphere; yet they are also more refined, since they are fibered knots. Here we begin our investigation of finite cyclic actions on simple fibered knots (S 2n+ι , K 2n~{) of dimension n > 3. Recall that a high dimensional knot is simple if its complement has the homotopy type of S ι up to but not including its middle dimension. In particular, we consider simple fibered knots for which the submanifold K is a rational homology sphere. The more general situation, which requires modification of the proofs and techniques given here, as well as the introduction of some further invariants will be discussed in a separate paper [15]. We consider both the free and the semifree cases. We obtain a classification of both types of actions, as well as a determination of the number theoretic conditions which guarantee their existence. We say that (S, K) admits a free Z m action if Z m acts freely on S leaving K invariant; we say that (S, K) admits a semifree action if the action on S is semifree with fixed set precisely K. Our results mirror those concerning spherical knots, found in [8], [13], [14], and [17], reflecting the fact that the objects of study are a generalization of these; the methods of proof necessarily address the nonvanishing of the homology of K and exploit the existence of the fibration of the complement.
Crossings and writhe of flexible and ideal knots
Physical Review E, 2001
The data of ideal knots ͓Nature, 384, 142 ͑1996͔͒ are reanalyzed and the average crossing number of the ideal knots ͗X͘ ideal shows a nonlinear behavior with the essential crossing number C. Supplemented with our Monte Carlo simulations using the bond fluctuation model on flexible knotted polymers, our analysis indicates that ͗X͘ ideal varies nonlinearly with both C and the corresponding average crossing number of the flexible knot, which is contrary to previous claims. Our extensive simulation data on the average crossing number of flexible knots suggest that it varies linearly with the square root of C. Furthermore, our data on the average writhe number ͗Wr͘ indicate that various knots are classified into holonomous groups, and ͗Wr͘ has a quantized linear increment with C in all four knot groups in our study.
Knot Invariants and Their Implications for Closed Plane Cur Ves
2005
A dominating feature of knot theory is the problem of knot classification. In this paper, we hope to simplify the task of classification through strong connections between Arnold's work with knot invariants and that of Xiao-Song Lin and Zhenghan Wang. We show that the defect of a plane curve associated with a flattened alternating knot may be considered an invariant for alternating knots. We also give simple method of computing a certain Vassiliev knot invariant from Arnold's plane curve invariants and signed crossings.
A Euclidean Geometric Invariant of Framed (Un)Knots in Manifolds
2006
We present an invariant of a three-dimensional manifold with a framed knot in it based on the Reidemeister torsion of an acyclic complex of Euclidean geometric origin. To show its nontriviality, we calculate the invariant for some framed (un)knots in lens spaces. Our invariant is related to a finite-dimensional fermionic topological quantum field theory.
Casson's invariant and surgery on knots
Proceedings of the Edinburgh Mathematical Society, 1992
We show that given a knot in a homology sphere there is a sequence of invariants with the property that if the nth invariant does not vanish, then this implies the existence of a family of irreducible representations of the fundamental group of the complement of the knot into SU(n).
On Ahlfors’ Schwarzian derivative and knots
Pacific Journal of Mathematics, 2007
We extend Ahlfors' definition of the Schwarzian derivative for curves in euclidean space to include curves on arbitrary manifolds, and give applications to the classical spaces of constant curvature. We also derive in terms of the Schwarzian a sharp criterion for a closed curve in ޒ 3 to be unknotted.
Journal of Differential Geometry
This paper studies knots that are transversal to the standard contact structure in R 3 , bringing techniques from topological knot theory to bear on their transversal classification. We say that a transversal knot type T K is transversally simple if it is determined by its topological knot type K and its Bennequin number. The main theorem asserts that any T K whose associated K satisfies a condition that we call exchange reducibility is transversally simple. As a first application, we prove that the unlink is transversally simple, extending the main theorem in [10]. As a second application we use a new theorem of Menasco [17] to extend a result of Etnyre [11] to prove that all iterated torus knots are transversally simple. We also give a formula for their maximum Bennequin number. We show that the concept of exchange reducibility is the simplest of the constraints that one can place on K in order to prove that any associated T K is transversally simple. We also give examples of pairs of transversal knots that we conjecture are not transversally simple.
A REMARK ON RASMUSSEN'S INVARIANT OF KNOTS
Journal of Knot Theory and Its Ramifications, 2007
We show that Rasmussen's invariant of knots, which is derived from Lee's variant of Khovanov homology, is equal to an analogous invariant derived from certain other filtered link homologies.
Holomorphic disks and knot invariants
Advances in Mathematics, 2004
We define a Floer-homology invariant for knots in an oriented threemanifold, closely related to the Heegaard Floer homologies for three-manifolds defined in [18]. We set up basic properties of these invariants, including an Euler characteristic calculation, behaviour under connected sums. Then, we establish a relationship with HF + for surgeries along the knot. Applications include calculation of HF + of threemanifolds obtained by surgeries on some special knots in S 3 , and also calculation of HF + for certain simple three-manifolds which fiber over the circle.