FINITE-TYPE KNOT INVARIANTS BASED ON THE BAND-PASS AND DOUBLED-DELTA MOVES

In his two pioneering articles [9, 10] Jerry Levine introduced and completely determined the algebraic concordance groups of odd dimensional knots. He did so by defining a host of invariants of algebraic concordance which he showed were a complete set of invariants. While being very powerful, these invariants are in practice often hard to determine, especially for knots with Alexander polynomials of high degree. We thus propose the study of a weaker set of invariants of algebraic concordance—the rational Witt classes of knots. Though these are rather weaker invariants than those defined by Levine, they have the advantage of lending themselves to quite manageable computability. We illustrate this point by computing the rational Witt classes of all pretzel knots. We give many examples and provide applications to obstructing sliceness for pretzel knots. Also, we obtain explicit formulae for the determinants and signatures of all pretzel knots. This article is dedicated to Jerry Levine and his lasting mathematical legacy; on the occasion of the conference “Fifty years since Milnor and Fox” held at Brandeis University on June 2–5, 2008.

Finite-type invariants of classical and virtual knots

Grope cobordism of classical knots

We introduce and study so-called self-indexed graphs. These are (oriented) finite graphs endowed with a map from the set of edges to the set of vertices. Such graphs naturally arise from classical knot and link diagrams. In fact, the graphs resulting from link diagrams have an additional structure, an integral flow. We call a self-indexed graph with integral flow a comte. The analogy with links allows us to define transformations of comtes generalizing the Reidemeister moves on link diagrams. We show that many invariants of links can be generalized to comtes, most notably the linking number, the Alexander polynomials, the link group, etc. We also discuss finite type invariants and quandle cocycle invariants of comtes.

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 $\Sigma \times [0,1]$, where $\Sigma$ is closed and oriented. We construct new concordance invariants by adapting the Chen-Milnor theory of links in $S^3$ to an extension of the group of a virtual knot. A key ingredient is the Bar-Natan $\textit{Zh}$ map, which allows for a geometric interpretation of the group extension. The group extension itself was originally defined by Silver-Williams. Our extended $\bar{\mu}$-invariants obstruct concordance to homologically trivial knots in thickened surfaces. We use them to give new examples of non-slice virtual knots having trivial Rasmussen invariant, graded genus, affine index (or writhe) polynomial, and generalized Alexander polynomial. Furthermore, we complete the slice status classification of all virtual knots up to five classical crossings and reduce to four (out of 92800) the number of virtual knots up to six classical crossings having unknown slice status. Our main application is to Turaev's concordance group $\mathscr{VC}$ of long knots on surfaces. Boden and Nagel proved that the concordance group $\mathscr{C}$ of classical knots in $S^3$ embeds into the center of $\mathscr{VC}$. In contrast to the classical knot concordance group, we show $\mathscr{VC}$ is not abelian; answering a question posed by Turaev.

Introduced in S. Nelson and M. Sakurai [Biquandle arrow weight enhancements, Internat. J. Math. 34(8) (2023) 2350046], biquandle arrow weight invariants are enhancements of the biquandle counting invariant for oriented virtual and classical knots defined from biquandle-colored Gauss diagrams using a tensor over an abelian group satisfying certain properties. In this paper, we categorify the biquandle arrow weight polynomial invariant using biquandle coloring quivers, obtaining new infinite families of polynomial invariants of oriented virtual and classical knots.

Knot theory is a branch of the geometric topology, the core question of knot theory is to explore the equivalence classification of knots; In other words, for a knot, how to determine whether the knot is an unknot; giving two knots, how to determine whether the two knots are equivalent. To prove that two knots are equivalent, it is necessary to turn one knot into another through the same mark transformation, but to show that two knots are unequal, the problem is not as simple as people think. We cannot say that they are unequal because we can't see the deformation between them. For the equivalence classification problem of knots, we mainly find equivalent invariants between knots. Currently, scholars have also defined multiple knot invariants, but they also have certain limitations, and even more difficult to understand. In this paper, based on existing theoretical results, we define a knot invariant through the skein relation with two equations. To prove this knot invariant, we define a function <i>f</i>(<I>L</I>), and to prove <i>f</i>(<I>L</I>) to be a homology invariant of a non-directed link, we need to show that it remains constant under the Reideminster moves. This article first defines the <i>f_k</i>(<I>L</I>), the property of <i>f</i>(<I>L</I>) is obtained by using the properties of <i>f_k</i>(<I>L</I>). In the process of proof, the induction method has been used many times. The proof process is somewhat complicated, but it is easier to understand. And the common knot invariant is defined by one equation, which defining the knot invariant with two equations in this paper.

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.

A longstanding open question in low dimensional topology was raised by J. H. C. Whitehead in 1941 [9]: “Is any subcomplex of an aspherical, two-dimensional complex itself aspherical?” The asphericity of classical knot complements [7] provides evidence that the answer to Whitehead's question might be “yes”. Indeed, each classical knot complement has the homotopy type of a two-complex which can be embedded in a finite contractible two-complex. This property is shared by a large class of four-manifolds; these are the ribbon disc complements, whose asphericity has been conjectured, and even claimed, but never proven. (See [4] for a discussion.) It is reasonable and convenient to formulate the following.

We define A k -moves for embeddings of a finite graph into the 3-sphere for each natural number k. Let A k -equivalence denote an equivalence relation generated by A k -moves and ambient isotopy. A k -equivalence implies A k-1 -equivalence. Let F be an A k-1 -equivalence class of the embeddings of a finite graph into the 3-sphere. Let g be the quotient set of F under A k -equivalence. We show that the set g forms an abelian group under a certain geometric operation. We define finite type invariants on F of order (n; k). And we show that if any finite type invariant of order (1; k) takes the same value on two elements of F, then they are A k -equivalent. A k -move is a generalization of C k -move defined by K. Habiro. Habiro showed that two oriented knots are the same up to C k -move and ambient isotopy if and only if any Vassiliev invariant of order < k - 1 takes the same value on them. The 'if' part does not hold for two-component links. Our result gives a sufficient condition for spatial graphs to be C k -equivalent.

Using Yoshikawa's surface diagram, we constructed new invariants of ambient isotopy classes of smoothly embedded closed surfaces in ℝ4 via a state-sum model similar to the Kauffman's state-sum model for the Jones polynomial for classical knots and links in ℝ3. It is shown that the invariants can also be defined by skein relation and thus they are calculated from Yoshikawa's surface diagrams recurrently. Some of the properties of the invariants are given and explicit computations for several surfaces are included.

The cubic lattice stick index of a knot type is the least number of sticks glued end-to-end that are necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p, p + 1)-torus knots, and show how composing and taking satellites can be used to obtain the cubic lattice stick index for a relatively large infinite class of knots. Additionally, we present several bounds relating cubic lattice stick index to other known invariants.

Let be a torus embedded in *S and X its exterior. The peripheral subgroup of is the image of nt(dX) by the map /„ induced by the inclusion i: dXc+X. So it is isomorphic to the direct sum of the infinite cyclic group Z, which is generated by a meridian, and some quotient of π1(F) = Z®Z. We denote by τF the second summand and call it the type of F see [10, §3]. If is unknotted, that is, bounds a solid torus in S 4 (see [7]), then τ = 0 . If is a torus constructed by spinning a non-trivial classical knot, then τF=Z, cf. [3, 10, 11]. Asano [1] and Litherland [10] constructed examples with τ = Z φ Z . And Boyle [3] showed that there are tori of type Zn for n = 2,5 and 10 by attaching a 2-dimensional 1-handle to the 5-twist-spun trefoil. We abbreviate the group π^S^ — F) of as πF. It is known that the second homology of the group H2(nF) is a quotient of H2(S ' — F) = Z φ Z, and several authors gave examples having non-trivial second homology [2, 4], see also [12]. Litherland [10] showed that H2(πF) is a quotient of τF and that any quotient of Z φ Z is realizable as H2{πF) for some torus of type Z φ Z . For abelian groups A and JB, we write A <B if A is a quotient of B. It is natural to ask

A conjecture of Shumakovitch states that every nontrivial knot has 2-torsion in its Khovanov homology.We show that if a knot K has no 2-torsion in its Khovanov homology, then the rank of its reduced Khovanov homology is minimal among all knots obtainable from K by a proper rational tangle replacement.It follows, for example, that unknotting number 1 knots have 2-torsion in their Khovanov homology. 57K18Shumakovitch conjectured that every nontrivial knot has an element of order 2 in its Khovanov homology [10, Conjecture 1].The conjecture has been verified for some infinite families of knots (see, for example, [2; 9; 10]) and has withstood large computational searches.In this note, we provide topological evidence for the conjecture, and we verify the conjecture for a large class of knots that include all unknotting number 1 knots.

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.