arXiv subjects
Publications and source records attributed to Xiao-Song Lin.
A famous result of Bennequin states that for any braid representative of the unknot the Bennequin number is negative. We will extend this result to all n-trivial closed n-braids. This is a class of infinitely many knots closed under taking mirror images. Our proof relies on a non-standard parametrization of the Homfly polynomial.
We construct a cubical CW-complex CK(M^3) whose rational cohomology algebra contains Vassiliev invariants of knots in the 3-manifold M^3. We construct \bar{CK}(R^3) by attaching cells to CK(R^3) for every degenerate 1-singular and 2-singular knot, and we show that π_1(\bar{CK}(R^3))=1 and π_2(\bar{CK}(R^3))=Z. We give conditions for Vassiliev invariants to be nontrivial in cohomology. In particular, for R^3 we show that v_2 uniquely generates H^2(CK,D), where D is the subcomplex of degenerate singular knots. More generally, we show that any Vassiliev invariant coming from the Conway polynomial is nontrivial in cohomology. The cup product in H^*(CK) provides a new graded commutative algebra of Vassiliev invariants evaluated on ordered singular knots. We show how the cup product arises naturally from a cocommutative differential graded Hopf algebra of ordered chord diagrams.
By calculating the Fermat limit of certain q-Fermat functions, we get explicit surgery formulae for the second and third Ohtsuki invariants for homology 3-spheres. The surgery formula of the second Ohtsuki invariant λ_2, combined with an argument using the general theory of finite type invariants of homology 3-spheres, leads to a congruence relation between the first and second Ohtsuki invariants: λ_1=2λ_2 mod 24.
We show that the Vassiliev invariants of orders $\leq n$ of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group. As a consequence of this, we obtain that the Vassiliev invariants of the knot $K=\partial S$ are null-concordance obstructions of certain links that can be obtained from regular spines of S. We also discuss various generalizations of these results, and we conjecture a geometric characterization of knots whose invariants of all orders vanish.
We will develop various methods, some are of geometric nature and some are of algebraic nature, to detect the various achiralities of knots and links in $S^3$. For example, we show that the twisted Whitehead double of a knot is achiral if and only if the double is the unknot or the figure eight knot, and we show that all non-trivial links with $\leq9$ crossings are not achiral except the Borromean rings. A simple procedure for calculating the $η$-function is given in terms of a crossing change formula and its initial values.
The inclusion of the space of all knots of a prescribed writhe in a particular isotopy class into the space of all knots in that isotopy class is a weak homotopy equivalence.
A model of random walk on knot diagrams is used to study the Alexander polynomial and the colored Jones polynomial of knots. In this context, the inverse of the Alexander polynomial of a knot plays the role of an Ihara-Selberg zeta function of a directed weighted graph, counting with weights cycles of random walk on a 1-string link whose closure is the knot in question. The colored Jones polynomial then counts with weights families of ``self-avoiding'' cycles of random walk on the cabling of the 1-string link. As a consequence of such interpretations of the Alexander and colored Jones polynomials, the computation of the limit of the renormalized colored Jones polynomial when the coloring (or cabling) parameter tends to infinity whereas the weight parameter tends to 1 leads immediately to a new proof of the Melvin-Morton conjecture, which was first established by Rozansky and by Bar-Natan and Garoufalidis.
We show that for an $n$-component, $n$-bridge link and a positive integer $m$, the following is true: If the longitudes of $L$ lie in the $(m+2)$-th term of the lower central series of the link group then all the finite type invariants of orders $\leq m$ for $L$ are the same as these of the $n$-component unlink.
Using a probabilistic interpretation of the Burau representation of the braid group offered by Vaughan Jones, we generalize the Burau representation to a representation of the semigroup of string links. This representation is determined by a linear system, and is dominated by finite type string link invariants. For positive string links, the representation matrix can be interpreted as the transition matrix of a Markov process. For positive non-separable links, we show that all states are persistent. (This is an extensively revised version of the first author's original paper titled "Burau representation and random walk on knots".)
We show that a variation of Milnor's $\barμ$-invariants, the so-called Campbell-Hausdorff invariants introduced recently by Stefan Papadima, are of finite type with respect to {\it marked singular links}. These link invariants are stronger than quantum invariants in the sense that they detect easily the non-invertibility of links with more than one components. It is still open whether some effectively computable knot invariants, e.g. finite type knot invariants (Vassiliev invariants), could detect the non-invertibility of knots.
This an article about some elementary geometric and combinatorial natures of various knot energies. A related "new" knot invariant -- the X-crossing number -- is introduced.
An attempt is made to conceptualize the derivation as well as to facilitate the computation of Ohtsuki's rational invariants $λ_n$ of integral homology 3-spheres extracted from Reshetikhin-Turaev SU(2) quantum invariants. Several interesting consequences will follow from our computation of $λ_2$. One of them says that $λ_2$ is always an integer divisible by 3. It seems interesting to compare this result with the fact shown by Murakami that $λ_{1}$ is 6 times the Casson invariant. Other consequences include some general criteria for distinguishing homology 3-spheres obtained from surgery on knots by using the Jones polynomial.
This is a survey on the current status of the study of finite type invariants of integral homology 3-spheres based on lectures given in the workshop on knot theory at Banach International Center of Mathematics, Warsaw, July 1995. As a new result, we show that the space of finite type invariants of integral homology 3-spheres is a graded polynomial algebra generated by invariants additive under the connected sum. We also discuss some open questions on this subject.
We construct a polynomial invariant, for links in a Seifert fibered or atoroidal rational homology 3-sphere, which generalizes the 2-variable Jones polynomial (HOMFLY). As a consequence, we show that the dual of the HOMFLY skein module of a homotopy 3-sphere is isomorpic to that of the genuine 3-sphere .
We prove that the knot invariant induced by a $\Bbb Z$-homology 3-sphere invariant of order $\leq k$ in Ohtsuki's sense, where $k\geq 4$, is of order $\leq k-2$. The method developed in our computation shows that there is no $\Bbb Z$-homology 3-sphere invariant of order 5. This result agrees with a conjecture of Rozansky based on physical predictions about the asymptotic behavior of Witten's Chern-Simons path integral.
We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves and it is related with invariants of generic plane curves defined by Arnold recently with deep motivations in symplectic and contact geometry. Quadratic bounds on these plane curve invariants are derived using their relationship with the knot invariant.