N-component Link Research Articles

Multi-boundary entanglement in Chern-Simons theory and link invariants

We consider Chern-Simons theory for gauge group G at level k on 3-manifolds Mn with boundary consisting of n topologically linked tori. The Euclidean path integral on Mn defines a quantum state on the boundary, in the n-fold tensor product of the torus Hilbert space. We focus on the case where Mn is the link-complement of some n-component link inside the three-sphere S3. The entanglement entropies of the resulting states define framing-independent link invariants which are sensitive to the topology of the chosen link. For the Abelian theory at level k (G = U(1)k) we give a general formula for the entanglement entropy associated to an arbitrary (m|n − m) partition of a generic n-component link into sub-links. The formula involves the number of solutions to certain Diophantine equations with coefficients related to the Gauss linking numbers (mod k) between the two sublinks. This formula connects simple concepts in quantum information theory, knot theory, and number theory, and shows that entanglement entropy between sublinks vanishes if and only if they have zero Gauss linking (mod k). For G = SU(2)k, we study various two and three component links. We show that the 2-component Hopf link is maximally entangled, and hence analogous to a Bell pair, and that the Whitehead link, which has zero Gauss linking, nevertheless has entanglement entropy. Finally, we show that the Borromean rings have a “W-like” entanglement structure (i.e., tracing out one torus does not lead to a separable state), and give examples of other 3-component links which have “GHZ-like” entanglement (i.e., tracing out one torus does lead to a separable state).

Local moves for links with common sublinks

A Ck-move is a local move that involves k+1 strands of a link. A Ck-move is called a Ckd-move if these k+1 strands belong to mutually distinct components of a link. Since a Ckd-move preserves all k-component sublinks of a link, we consider the converse implication: are two links with common k-component sublinks related by a sequence of Ckd-moves? We show that the answer is yes under certain assumptions, and provide explicit counter-examples for more general situations. In particular, we consider (n,k)-Brunnian links, i.e. n-component links whose k-component sublinks are all trivial. We show that such links can be deformed into a trivial link by Ckd-moves, thus generalizing a result of Habiro and Miyazawa–Yasuhara, and deduce some results on finite type invariants of (n,k)-Brunnian links.

Milnor invariants and the HOMFLYPT Polynomial

We give formulas expressing Milnor invariants of an n-component link L in the 3-sphere in terms of the HOMFLYPT polynomial as follows. If the Milnor invariant \bar{\mu}_J(L) vanishes for any sequence J with length at most k, then any Milnor \bar{\mu}-invariant \bar{\mu}_I(L) with length between 3 and 2k+1 can be represented as a combination of HOMFLYPT polynomial of knots obtained from the link by certain band sum operations. In particular, the `first non vanishing' Milnor invariants can be always represented as such a linear combination.

ACHIRALITY AND LINKING NUMBERS OF LINKS

An oriented and ordered n-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring orientations and ordering of the components. For an oriented and ordered n-component link L, let λL be the product of linking numbers of all 2-component sublinks in L. For n = 4m + 3, where m is a non-negative integer, we show that if L is achiral then λL = 0. And for n ≠ 4m + 3, we show that there exists an n-component achiral link L with λL ≠ 0 by using achiral embeddings of complete graphs with n vertices Kn.

Vassiliev invariants of type one for links in the solid torus

In Bataineh (2003) [2] we studied the type one invariants for knots in the solid torus. In this research we study the type one invariants for n-component links in the solid torus by generalizing Aicardi's invariant for knots in the solid torus to n-component links in the solid torus. We show that the generalized Aicardi's invariant is the universal type one invariant, and we show that the generalized Aicardi's invariant restricted to n-component links in the solid torus with zero winding number for each component is equal to an invariant we define using the universal cover of the solid torus. We also define and study a geometric invariant for n-component links in the solid torus. We give a lower bound on this invariant using the type one invariants, which are easy to calculate, which helps in computing this geometric invariant, which is usually hard to calculate.

INTRINSICALLY n-LINKED COMPLETE BIPARTITE GRAPHS

We prove that every embedding of K2n+1,2n+1 into ℝ3 contains a non-split link of n components. Further, given an embedding of K2n+1,2n+1 in ℝ3, every edge of K2n+1,2n+1 is contained in a non-split n-component link in K2n+1,2n+1.

The integral cohomology of the group of loops

Let PSigma_n denote the group that can be thought of either as the group of motions of the trivial n-component link or the group of symmetric automorphisms of a free group of rank n. The integral cohomology ring of PSigma_n is determined, establishing a conjecture of Brownstein and Lee.

Intrinsically linked graphs and even linking number

We study intrinsically linked graphs where we require that every embedding of the graph contains not just a non-split link, but a link that satisfies some additional property. Examples of properties we address in this paper are: a two component link with lk(A,L) = k2^r, k not 0, a non-split n-component link where all linking numbers are even, or an n-component link with components L, A_i where lk(L,A_i) = 3k, k not 0. Links with other properties are considered as well. For a given property, we prove that every embedding of a certain complete graph contains a link with that property. The size of the complete graph is determined by the property in question.

Classification of n-component Brunnian links up to [formula omitted]-move

We give a classification of n-component links up to C n -move. In order to prove this classification, we characterize Brunnian links, and have that a Brunnian link is ambient isotopic to a band sum of a trivial link and Milnor's links.

INTRINSICALLY n-LINKED GRAPHS

For every natural number n, we exhibit a graph with the property that every embedding of it in ℝ3 contains a non-split n-component link. Furthermore, we prove that our graph is minor minimal in the sense that every minor of it has an embedding in ℝ3 that contains no non-split n-component link.

SPHERE-SLICE LINKS WITH AT MOST FIVE COMPONENTS

A sphere-slice link is a cross-sectional link obtained by cutting a 2-sphere in R4 by a hyperplane. We characterize a sphere-slice link with at most 5 components by linking numbers. We also show that every n-component link is sublink of a (2n+1)-component sphere-slice link and this value is best possible.

The Third Derivative of the Jones Polynomial

Let L be an oriented n-component link and let VL (t) be the Jones polynomial of L. A relationship between [Formula: see text] (1) and the coefficients of the Conway polynomials of L is given.

On a certain move generating link-homology

On the other hand, the first author defined a #-unknot t ing operation on an oriented link diagram as in Fig. 0.2 [11]. It is proved in [11] that every knot can be deformed into a trivial knot by a finite sequence of @-unknotting operations. It can be also proved (see Appendix) that for two oriented and ordered n-component links L = K1 u K 2 u . . . u K , and L' = K'~ uK~ ~. . . u K',, L can be deformed into E by a finite sequence of @-unknotting operations if and only if Ik(K~,L--Ki) ==-lk(K'i, L-K'i) (rood2) for every i (1 < i < n). (Here, lk denotes the linking number in S3.)

A new technique for the link slice problem

The conjectures that the 4-dimensional surgery theorem and 5-dimensional s-cobordism theorem hold without fundamental group restriction in the topological category are equivalent to assertions that certain "atomic" links are slice. This has been reported in [CF, F2, F4 and FQ]. The slices must be topologically flat and obey some side conditions. For surgery the condition is: ~a(S 3 ~ slice)--, rq (B 4 slice) must be an epimorphism, i.e., the slice should be "homotopically ribbon"; for the s-cobordism theorem the slice restricted to a certain trivial sublink must be standard. There is some choice about what the atomic links are; the current favorites are built from the simple "Hopf link" by a great deal of Bing doubling and just a little Whitehead doubling. A link typical of those atomic for surgery is illustrated in Fig. 1. (Links atomic for both s-cobordism and surgery are slightly less symmetrical.) There has been considerable interplay between the link theory and the equivalent abstract questions. The link theory has been of two sorts: algebraic invariants of finite links and the limiting geometry of infinitely iterated links. Our object here is to solve a class of free-group surgery problems, specifically, to construct certain slices for the class of links ~ where D(L)eCg if and only if D(L) is an untwisted Whitehead double of a boundary link L. Unfortunately, (~ appears not to be atomic for surgery or s-cobordism. However our technique is quite different from the earlier extension [F4] of simply-connected methods and may prove helpful in establishing the limits of the nonsimply-connected theory. An n-component link is a topological imbedding I ) Si c L , $3 of n circles i = 1 in the 3-sphere. It is slice if and only if there is a topological imbedding L (dotted arrow) making the diagram: