Local moves on spatial graphs and finite type invariants

23 pages

Grope cobordism of classical knots

In his study of the group of homology cylinders, J. Levine made the conjecture that a certain homomorphism eta': T -> D' is an isomorphism. Here T is an abelian group on labeled oriented trees, and D' is the kernel of a bracketing map on a quasi-Lie algebra. Both T and D' have strong connections to a variety of topological settings, including the mapping class group, homology cylinders, finite type invariants, Whitney tower intersection theory, and the homology of the group of automorphisms of the free group. In this paper, we confirm Levine's conjecture. This is a central step in classifying the structure of links up to grope and Whitney tower concordance, as explained in other papers of this series. We also confirm and improve upon Levine's conjectured relation between two filtrations of the group of homology cylinders.

Motivated by Arnold's theory of invariants of plane curves, we introduce the semi-group of equivalence classes of arrangements of nested curves. There exists a natural invariant of plane curves without inverse self-tangencies with values in this semi-group. We show that the associated Grothendieck group is ℤ × ℤ. These two factors correspond to previously known invariants of plane curves without inverse self-tangencies, namely Whitney's index and Arnold's J- invariant. We show that arrangements of nested curves are not classified by their finite type invariants.

On the Vassiliev knot invariants

We describe an efficient algorithm to compute finite type invariants of type k k by first creating, for a given knot K K with n n crossings, a look-up table for all subdiagrams of K K of size ⌈ k 2 ⌉ \lceil \frac {k}{2}\rceil indexed by dyadic intervals in [ 0 , 2 n − 1 ] [0,2n-1] . Using this algorithm, any such finite type invariant can be computed on an n n -crossing knot in time O ~ ( n ⌈ k 2 ⌉ ) {\tilde {O}}(n^{\lceil \frac {k}{2}\rceil }) , a lot faster than the previously best published bound of O ~ ( n k ) {\tilde {O}}(n^k) .

We explain a direct topological proof for the multiplicativity of Duflo\nisomorphism for arbitrary finite dimensional Lie algebras, and derive the\nexplicit formula for the Duflo map. The proof follows a series of implications,\nstarting with "the calculation 1+1=2 on a 4D abacus", using the study of\nhomomorphic expansions (aka universal finite type invariants) for ribbon\n2-knots, and the relationship between the corresponding associated graded space\nof arrow diagrams and universal enveloping algebras. This complements the\nresults of the first author, Le and Thurston, where similar arguments using a\n"3D abacus" and the Kontsevich Integral were used to derive Duflo's theorem for\nmetrized Lie algebras; and results of the first two authors on finite type\ninvariants of w-knotted objects, which also imply a relation of 2-knots with\nDuflo's theorem in full generality, though via a lengthier path.\n

We introduce a homology surgery problem in dimension 3 which has the property that the vanishing of its algebraic obstruction leads to a canonical class of \pi-algebraically-split links in 3-manifolds with fundamental group \pi . Using this class of links, we define a theory of finite type invariants of 3-manifolds in such a way that invariants of degree 0 are precisely those of conventional algebraic topology and surgery theory. When finite type invariants are reformulated in terms of clovers, we deduce upper bounds for the number of invariants in terms of \pi-decorated trivalent graphs. We also consider an associated notion of surgery equivalence of \pi-algebraically split links and prove a classification theorem using a generalization of Milnor's \mu-invariants to this class of links.

In this paper, we study the finite type invariants of Gauss words. In the Polyak algebra techniques, we reduce the determination of the group structure to transformation of a matrix into its Smith normal form and we give the simplified form of a universal finite type invariant by means of the isomorphism of this transformation. The advantage of this process is that we can implement it as a computer program. We obtain the universal finite type invariant of degrees 4, 5 and 6 explicitly. Moreover, as an application, we give the complete classification of Gauss words of rank 4 and the partial classification of Gauss words of rank 5 where the distinction of only one pair remains.

We study a theory of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries. It is an analogue in the setting of the rational homology of the Goussarov-Rozansky theory for knots in integral homology 3-spheres. We give a partial combinatorial description of the graded space associated with our theory and determine some cases when this description is complete. For null-homologous knots in rational homology 3-spheres with a trivial Alexander polynomial, we show that the Kricker lift of the Kontsevich integral and the Lescop equivariant invariant built from integrals in configuration spaces are universal finite type invariants for this theory; in particular it implies that they are equivalent for such knots.

Finite-type invariants of classical and virtual knots

By classical results of Rochlin, Thom, Wallace and Lickorish, it is well-known that any two 3-manifolds (with diffeomorphic boundaries) are related one to the other by surgery operations. Yet, by restricting the type of the surgeries, one can define several families of non-trivial equivalence relations on the sets of (diffeomorphism classes of) 3-manifolds. In this expository paper, which is based on lectures given at the school “Winter Braids XI” (Dijon, December 2021), we explain how certain filtrations of mapping class groups of surfaces enter into the definitions and the mutual comparison of these surgery equivalence relations. We also survey the ways in which concrete invariants of 3-manifolds (such as finite-type invariants) can be used to characterize such relations.

We study generalizations of finite-type knot invariants obtained by replacing the crossing change in the Vassiliev skein relation by some other local move, analyzing in detail the band-pass and doubled-delta moves. Using braid-theoretic techniques, we show that, for a large class of local moves, generalized Goussarov's n-equivalence classes of knots form groups under connected sum. (Similar results, but with a different approach, have been obtained before by Taniyama and Yasuhara.) It turns out that primitive band-pass finite-type invariants essentially coincide with standard primitive finite-type invariants, but things are more interesting for the doubled-delta move. The complete degree 0 doubled-delta invariant is the S-equivalence class of the knot. In this context, we generalize a result of Murakami and Ohtsuki to show that the only primitive Vassiliev invariants of S-equivalence taking values in an abelian group with no 2-torsion arise from the Alexander–Conway polynomial. We start analyzing degree one doubled-delta invariants by considering which Vassiliev invariants are of doubled-delta degree one, finding that there is exactly one such invariant in each odd Vassiliev degree, and at most one (which is ℤ2-valued) in each even Vassiliev degree. Analyzing higher doubled-delta degrees, we observe that the Euler degree n + 1 part of Garoufalidis and Kricker's rational lift of the Kontsevich integral is a doubled-delta degree 2n invariant.

A local move called a Cn-move is closely related to Vassiliev invariants. The Cn-distance between two knots K and L, denoted by dCn(K, L), is the minimal number of Cn-moves needed to transform K into L. In the case of n ≥ 3, we show that for any pair of knots K1and K2with dCn(K1, K2) = 1 and for any given natural number m, there exist infinitely many knots Jj(j = 1, 2, …) such that dCn(K1, Jj) = dCn(Jj, K2) = 1 and they have the same Vassiliev invariants of order less than or equal to m.

Physics-informed spatio-temporal hybrid neural networks for predicting remaining useful life in aircraft engine