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Xiao-Song Lin

Publications and source records attributed to Xiao-Song Lin.

At least 19 recordsLinked to original sources
arXiv2007

Alternating sum formulae for the determinant and other link invariants

A classical result states that the determinant of an alternating link is equal to the number of spanning trees in a checkerboard graph of an alternating connected projection of the link. We generalize this result to show that the determinant is the alternating sum of the number of quasi-trees of genus j of the dessin of a non-alternating link. Furthermore, we obtain formulas for other link invariants by counting quantities on dessins. In particular we will show that the $j$-th coefficient of the Jones polynomial is given by sub-dessins of genus less or equal to $j$.

math.GT↗
arXiv2007

Seifert surfaces, Commutators and Vassiliev invariants

We show that the Vassiliev invariants 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.

math.GT↗
arXiv2007

The Jones polynomial and graphs on surfaces

The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobas-Riordan-Tutte polynomial generalizes the Tutte polynomial of planar graphs to graphs that are embedded in closed oriented surfaces of higher genus. In this paper we show that the Jones polynomial of any link can be obtained from the Bollobas-Riordan-Tutte polynomial of a certain oriented ribbon graph associated to a link projection. We give some applications of this approach.

math.GT↗
arXiv2006

On Generalized Knot Groups

Generalized knot groups G_n(K) were introduced first by Wada and Kelly independently. The classical knot group is the first one G_1(K) in this series of finitely presented groups. For each natural number n, G_1(K) is a subgroup of G_n(K) so the generalized knot groups can be thought of as extensions of the classical knot group. For the square knot SK and the granny knot GK, we have an isomorphism $G_1(SK)\cong G_1(GK)$. From the presentations of G_n(SK) and G_n(GK), for n>1, it seems unlikely that G_n(SK) and G_n(GK) would be isomorphic to each other. We are able to show that for many finite groups H, the numbers of homomorphisms from G_n(SK) and G_n(GK) to H, respectively, are the same. Moreover, the numbers of conjugacy classes of homomorphisms from G_n(SK) and G_n(GK) to H, respectively, are also the same. It remains a challenge to us to show, as we would like to conjecture, that G_n(SK) and G_n(GK) are not isomorphic to each other for all n>1.

math.GT↗
arXiv2006

Knot adjacency, genus and essential tori

A knot K is called n-adjacent to another knot K', if K admits a projection containing n generalized crossings such that changing any 0 < m \leq n of them yields a projection of K'. We apply techniques from the theory of sutured 3-manifolds, Dehn surgery and the theory of geometric structures of 3-manifolds to answer the question of the extent to which non-isotopic knots can be adjacent to each other. A consequence of our main result is that if K is n-adjacent to K' for all n, then K and K' are isotopic. This provides a partial verification of the conjecture of V. Vassiliev that the finite type knot invariants distinguish all knots. We also show that if no twist about a crossing circle L of a knot K changes the isotopy class of K, then L bounds a disc in the complement of K. This gives a characterization of the nugatory crossings of a knot.

math.GT↗
arXiv2006

Knot adjacency and fibering

It is known that the Alexander polynomial detects fibered knots and 3-manifolds that fiber over the circle. In this note, we show that when the Alexander polynomial becomes inconclusive, the notion of "knot adjacency", studied in the paper "Knot adjacency, genus and essential tori" by the authors, can be used to obtain obstructions to fibering of knots and of 3-manifolds. As an application, given a fibered knot K', we construct infinitely many non-fibered knots that share the same Alexander module and the same Vassiliev invariants up to certain orders with K'. Our construction also provides, for every natural number n, examples of irreducible 3-manifolds that cannot be distinguished by the Cochran-Melvin finite type invariants of order < n.

math.GT↗
arXiv2006

Torsion of Quasi-Isomorphisms

In this paper, we introduce the notion of Reidemeister torsion for quasi-isomorphisms of based chain complexes over a field. We call a chain map a quasi-isomorphism if its induced homomorphism between homology is an isomorphism. Our notion of torsion generalizes the torsion of acyclic based chain complexes, and is a chain homotopy invariant on the collection of all quasi-isomorphisms from a based chain complex to another. It shares nice properties with torsion of acyclic based chain complexes, like multiplicativity and duality. We will further generalize our torsion to quasi-isomorphisms between free chain complexes over a ring under some mild condition. We anticipate that the study of torsion of quasi-isomorphisms will be fruitful in many directions, and in particular, in the study of links in 3-manifolds.

math.AT↗
arXiv2006

On the Head and the Tail of the Colored Jones Polynomial

The colored Jones polynomial is a series of one variable Laurent polynomials J(K,n) associated with a knot K in 3-space. We will show that for an alternating knot K the absolute values of the first and the last three leading coefficients of J(K,n) are independent of n when n is sufficiently large. Computation of sample knots indicates that this should be true for any fixed leading coefficient of the colored Jones polynomial for alternating knots. As a corollary we get a Volume-ish Theorem for the colored Jones Polynomial.

math.GT↗
arXiv2006

On the Hecke algebras and the colored HOMFLY polynomial

The colored HOMFLY polynomial is the quantum invariant of oriented links in $S^3$ associated with irreducible representations of the quantum group $U_q(\mathrm{sl}_N)$. In this paper, using an approach to calculate quantum invariants of links via cabling-projection rule, we derive a formula for the colored HOMFLY polynomial in terms of the characters of the Hecke algebras and Schur polynomials. The technique leads to a fairly simple formula for the colored HOMFLY polynomial of torus links. This formula allows us to test the Labastida-Mari\~no-Vafa conjecture, which reveals a deep relationship between Chern-Simons gauge theory and string theory, on torus links.

math.QA↗
arXiv2005

A New Way to Tabulate Knots

We introduce a new way to tabulate knots by representing knot diagrams using a pair of planar trees. This pair of trees have their edges labeled by integers, they have no valence 2 vertices, and they have the same number of valence 1 vertices. The number of valence 1 vertices of the trees is called the girth of the knot diagram. The classification problem of knots admitting girth 2 and 3 diagrams is studied. The planar tree pair representations of girth less than 4 for knot diagrams in Rolfsen's table are given.

math.GT↗
arXiv2005

Colored Coalescent Theory

We introduce a colored coalescent process which recovers random colored genealogical trees. Here a colored genealogical tree has its vertices colored black or white. Moving backward along the colored genealogical tree, the color of vertices may change only when two vertice coalesce. The rule that governs the change of color involves a parameter $x$. When $x=1/2$, the colored coalescent process can be derived from a variant of the Wright-Fisher model for a haploid population in population genetics. Explicit computations of the expectation and the cumulative distribution function of the coalescent time are carried out. For example, our calculation shows that when $x=1/2$, for a sample of $n$ colored individuals, the expected time for the colored coalescent process to reach a black MRAC or a white MRAC, respectively, is $3-2/n$. On the other hand, the expected time for the colored coalescent process to reach a MRAC, either black or white, is $2-2/n$, which is the same as that for the standard Kingman coalescent process.

math.PR↗
arXiv2005

On n-punctured ball tangles

We consider a class of topological objects in the 3-sphere $S^3$ which will be called {\it $n$-punctured ball tangles}. Using the Kauffman bracket at $A=e^{πi/4}$, an invariant for a special type of $n$-punctured ball tangles is defined. The invariant $F$ takes values in $PM_{2\times2^n}(\mathbb Z)$, that is the set of $2\times 2^n$ matrices over $\mathbb Z$ modulo the scalar multiplication of $\pm1$. This invariant leads to a generalization of a theorem of D. Krebes which gives a necessary condition for a given collection of tangles to be embedded in a link in $S^3$ disjointly. We also address the question of whether the invariant $F$ is surjective onto $PM_{2\times2^n}(\mathbb Z)$. We will show that the invariant $F$ is surjective when $n=0$. When $n=1$, $n$-punctured ball tangles will also be called spherical tangles. We show that $\text{det} F(S)=0$ or 1 {\rm mod} 4 for every spherical tangle $S$. Thus $F$ is not surjective when $n=1$.

math.GT↗
arXiv2004

Continuous Time Markov Processes on Graphs

We study continuous time Markov processes on graphs. The notion of frequency is introduced, which serves well as a scaling factor between any Markov time of a continuous time Markov process and that of its jump chain. As an application, we study ``multi-person simple random walks'' on a graph G with n vertices. There are n persons distributed randomly at the vertices of G. In each step of this discrete time Markov process, we randomly pick up a person and move it to a random adjacent vertex. We give estimate on the expected number of steps for these $n$ persons to meet all together at a specific vertex, given that they are at different vertices at the begininng. For regular graphs, our estimate is exact.

math.PR↗
arXiv2004

A volume-ish theorem for the Jones polynomial of alternating knots

The Volume conjecture claims that the hyperbolic Volume of a knot is determined by the colored Jones polynomial. The purpose of this article is to show a Volume-ish theorem for alternating knots in terms of the Jones polynomial, rather than the colored Jones polynomial: The ratio of the Volume and certain sums of coefficients of the Jones polynomial is bounded from above and from below by constants. Furthermore, we give experimental data on the relation of the growths of the hyperbolic volume and the coefficients of the Jones polynomial, both for alternating and non-alternating knots.

math.GT↗
arXiv2003

Knot adjacency and satellites

A knot K is called n-adjacent to the unknot, if K admits a projection containing n generalized crossings such that changing any m (no larger than n) of them yields a projection of the unknot. We show that a non-trivial satellite knot K is n-adjacent to the unknot, for some n>0, if and only if it is n-adjacent to the unknot in any companion solid torus. In particular, every model knot of K is n-adjacent to the unknot. Along the way of proving these results, we also show that 2-bridge knots of the form K_{p/q}, where p/q=[2q_1,2q_2] for some integers q_1,q_2, are precisely those knots that have genus one and are 2-adjacent to the unknot.

math.GT↗
arXiv2003

On the bridge number of knot diagrams with minimal crossings

Given a diagram $D$ of a knot $K$, we consider the number $c(D)$ of crossings and the number $b(D)$ of overpasses of $D$. We show that, if $D$ is a diagram of a nontrivial knot $K$ whose number $c(D)$ of crossings is minimal, then $1+\sqrt{1+c(D)} \leq b(D)\leq c(D)$. These inequalities are shape in the sense that the upper bound of $b(D)$ is achieved by alternating knots and the lower bound of $b(D)$ is achieved by torus knots. The second inequality becomes an equality only when the knot is an alternating knot. We prove that the first inequality becomes an equality only when the knot is a torus knot.

math.GT↗
arXiv2000

Link-homotopy Invariants of Finite Type

An explicit polynomial in the linking numbers $l_{ij}$ and Milnor's triple linking numbers $μ(rst)$ on six component links is shown to be a well-defined finite type link-homotopy invariant. This solves a problem raised by B. Mellor and D. Thurston. An extension of our construction also produces a finite type link invariant which detects the invertibility for some links.

math.GT↗