相关论文: Gordian distance and clasper surgery for links
A C_k-move is a local move that involves (k+1) strands of a link. A C_k-move is called a C_k^d-move if these (k+1) strands belong to mutually distinct components of a link. Since a C_k^d-move preserves all k-component sublinks of a link, we…
In 1993 K. Habiro defined $C_k$-move of oriented links and around 1994 he proved that two oriented knots are transformed into each other by $C_k$-moves if and only if they have the same Vassiliev invariants of order $\leq k-1$. In this…
This paper is expository and is accessible to students. We define simple invariants of knots or links (linking number, Arf-Casson invariants and Alexander-Conway polynomials) motivated by interesting results whose statements are accessible…
We use the knot homology of Khovanov and Lee to construct link concordance invariants generalizing the Rasmussen $s$-invariant of knots. The relevant invariant for a link is a filtration on a vector space of dimension $2^{|L|}$. The basic…
In this paper, we give a complete set of finite type string link invariants of degree <5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closure of…
A C_n-move is a local move on links defined by Habiro and Goussarov, which can be regarded as a `higher order crossing change'. We use Milnor invariants with repeating indices to provide several classification results for links up to…
We introduce the concept of `claspers,' which are surfaces in 3-manifolds with some additional structure on which surgery operations can be performed. Using claspers we define for each positive integer k an equivalence relation on links…
It has long been known that a Milnor invariant with no repeated index is an invariant of link homotopy. We show that Milnor's invariants with repeated indices are invariants not only of isotopy, but also of self C_k-moves. A self C_k-move…
The slicing number of a knot, $u_s(K)$, is the minimum number of crossing changes required to convert $K$ to a slice knot. This invariant is bounded above by the unknotting number and below by the slice genus $g_s(K)$. We show that for many…
J.P. Levine introduced a clover link to investigate the indeterminacy of the Milnor invariants of a link. It is shown that for a clover link, the Milnor numbers of length at most $2k+1$ are well-defined if those of length at most $k$…
In the 1980's Daryl Cooper introduced the notion of a C-complex (or clasp-complex) bounded by a link and explained how to compute signatures and polynomial invariants using a C-complex. Since then this was extended by works of Cimasoni,…
Symmetries of knots have been studied extensively, and strongly invertible knots are one of them. Lamm defined the equivariant crossing number $c_t(K)$, the minimum crossing number among all symmetric diagrams for a strongly invertible knot…
We construct geometrically two universal link invariants: universal ADO invariant and universal Jones invariant, as limits of invariants given by graded intersections in configuration spaces. More specifically, for a fixed level $\mathscr…
It is a natural question to ask whether two links are equivalent by the following moves -- parallel parts of a link are changed to k-times half-twisted parts and if they are, how many moves are needed to go from one link to the other. In…
Minor typographical errors fixed. Cochran constructed many links with Alexander module that of the unlink and some nonvanishing Milnor invariants, using as input commutators in a free group and as an invariant the longitudes of the links.…
In his 1957 paper, John Milnor introduced a collection of invariants for links in $S^3$ detecting higher-order linking phenomena by studying lower central quotients of link groups and comparing them to those of the unlink. These invariants,…
Introducing a way to modify knots using $n$-trivial rational tangles, we show that knots with given values of Vassiliev invariants of bounded degree can have arbitrary unknotting number (extending a recent result of Ohyama, Taniyama and…
Milnor's $\bar{\mu}$-invariants of links in the $3$-sphere $S^3$ vanish on any link concordant to a boundary link. In particular, they are trivial on any knot in $S^3$. Here we consider knots in thickened surfaces $\Sigma \times [0,1]$,…
Using unknotting number, we introduce a link diagram invariant of Hass and Nowik type, which changes at most by 2 under a Reidemeister move. As an application, we show that a certain infinite sequence of diagrams of the trivial…
Given a link in $S^3$ we will use invariants derived from the Alexander module and the Blanchfield pairing to obtain lower bounds on the Gordian distance between links, the unlinking number and various splitting numbers. These lower bounds…