162 citations · 348 across the 33 of their papers we have counts for
20 papers · 1 filter
Seifert surfaces, Commutators and Vassiliev invariants
Efstratia Kalfagianni, Xiao-Song Lin
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…
Alternating sum formulae for the determinant and other link invariants
Oliver T. Dasbach, David Futer, Efstratia Kalfagianni +2
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…
The Jones polynomial and graphs on surfaces
Oliver T. Dasbach, David Futer, Efstratia Kalfagianni +2
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 li…
On the Head and the Tail of the Colored Jones Polynomial
Oliver T. Dasbach, Xiao-Song Lin
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 va…
A New Way to Tabulate Knots
Lisa Hernandez, Xiao-Song Lin
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…
On n-punctured ball tangles
Jae-Wook Chung, Xiao-Song Lin
We consider a class of topological objects in the 3-sphere which will be called {\it -punctured ball tangles}. Using the Kauffman bracket at , an invariant for…
On Generalized Knot Groups
Xiao-Song Lin, Sam Nelson
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.…
Knot adjacency and fibering
Efstratia Kalfagianni, Xiao-Song Lin
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 inconcl…
Knot adjacency, genus and essential tori
Efstratia Kalfagianni, Xiao-Song Lin
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'.…
Knot adjacency and satellites
Efstratia Kalfagianni, Xiao-Song Lin
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…
On the bridge number of knot diagrams with minimal crossings
Jae-Wook Chung, Xiao-Song Lin
Given a diagram of a knot , we consider the number of crossings and the number of overpasses of . We show that, if is a diagram of a nontrivial knot …
Link-homotopy Invariants of Finite Type
Xiao-Song Lin
An explicit polynomial in the linking numbers and Milnor's triple linking numbers on six component links is shown to be a well-defined finite type link-homotopy i…
The Bennequin number of n-trivial closed n-braids is negative
Oliver T. Dasbach, 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. T…
Vassiliev invariants and the cubical knot complex
Ilya Kofman, Xiao-Song Lin
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 ce…
Achirality of knots and links
Boju Jiang, Xiao-Song Lin, Shicheng Wang +1
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 . For example, we show…
Subspaces of Knot Spaces
Craig Benham, Xiao-Song Lin, David Miller
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.
Random Walk on Knot Diagrams, Colored Jones Polynomial and Ihara-Selberg Zeta Function
Xiao-Song Lin, Zhenghan Wang
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 polynomia…
Fermat limit and congruence of Ohtsuki invariants
Xiao-Song Lin, Zhenghan Wang
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 form…
Milnor and finite type invariants of plat-closures
Efstratia Kalfagianni, Xiao-Song Lin
We show that for an -component, -bridge link and a positive integer , the following is true: If the longitudes of lie in the -th term of the lower central serie…
Regular Seifert surfaces and Vassiliev knot invariants
Efstratia Kalfagianni, Xiao-Song Lin
We show that the Vassiliev invariants of orders of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the compl…