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 knots, previous bounds on unknotting number obtained by Ozsvath and Szabo and by the author in fact give bounds on the slicing number. Livingston defined another invariant $U_s(K)$ which takes into account signs of crossings changed to get a slice knot, and which is bounded above by the slicing number and below by the slice genus. We exhibit an infinite family of knots $K_n$ with slice genus $n$ and Livingston invariant greater than $n$. Our bounds are based on restrictions (using Donaldson's diagonalisation theorem or Heegaard Floer homology) on the intersection forms of four-manifolds bounded by the double branched cover of a knot.
Given a knot complement X and its p-fold cyclic cover X_p, we identify twisted polynomials associated to 1-dimensional linear representations of the fundamental group of X_p with twisted polynomials associated to related p-dimensional linear representations of the fundamental group of X. This provides a simpler and faster algorithm to compute these twisted polynomials, allowing us to prove that 16 (of 18 previously unknown) algebraically slice knots of 12 or fewer crossings are not slice. We also use this improved algorithm to prove that the 24 mutants of the pretzel knot P(3,7,9,11,15), corre…
From Furuta's $\frac{10}{8}$ theorem, we derive a smooth slicing obstruction for knots in $S^3$ using a spin $4$-manifold whose boundary is $0$-surgery on a knot. We show that this obstruction is able to detect torsion elements in the smooth concordance group and find topologically slice knots which are not smoothly slice.
A knot in the 3-sphere is called doubly slice if it is a slice of an unknotted 2-sphere in the 4-sphere. We give a bi-sequence of new obstructions for a knot being doubly slice. We construct it following the idea of Cochran-Orr-Teichner's filtration of the classical knot concordance group. This yields a bi-filtration of the monoid of knots (under the connected sum operation) indexed by pairs of half integers. Doubly slice knots lie in the intersection of this bi-filtration. We construct examples of knots which illustrate non-triviality of this bi-filtration at all levels. In particular, these …
We apply Heegaard-Floer homology theory to establish generalized slicing Bennequin inequalities closely related to a recent result of T. Mrowka and Y. Rollin proved using Seiberg-Witten monopoles.
We study three knot invariants related to smoothly immersed disks in the four-ball. These are the four-ball crossing number, which is the minimal number of normal double points of such a disk bounded by a given knot; the slicing number, which is the minimal number of crossing changes to a slice knot; and the concordance unknotting number, which is the minimal unknotting number in a smooth concordance class. Using Heegaard Floer homology we obtain bounds that can be used to determine two of these invariants for all prime knots with crossing number ten or less, and to determine the concordance u…
We study the structure of the exteriors of gropes and Whitney towers in dimension 4, focusing on their fundamental groups. In particular we introduce a notion of unknottedness of gropes and Whitney towers in the 4-sphere. We prove that various modifications of gropes and Whitney towers preserve the unknottedness and do not enlarge the fundamental group. We exhibit handlebody structures of the exteriors of gropes and Whitney towers constructed by earlier methods of Cochran, Teichner, Horn, and the first author, and use them to construct examples of unknotted gropes and Whitney towers. As an app…
Let K be a knot in S^3. We study the iterated Bing doubles of K, giving a new proof for the following statement: If BD_n(K) is slice for some n, then K is algebraically slice. This result was first proved by Cha and Kim using covering link calculus. We also use this tool, but our proof is substantially simpler and illuminates several generalizations.
Bing doubling is an operation which produces a 2-component boundary link B(K) from a knot K. If K is slice, then B(K) is easily seen to be boundary slice. In this paper, we investigate whether the converse holds. Our main result is that if B(K) is boundary slice, then K is algebraically slice. We also show that the Rasmussen invariant can tell that certain Bing doubles are not smoothly slice.
The non-orientable 4-genus of a knot in the 3-sphere is defined as the smallest first Betti number of any non-orientable surface smoothly and properly embedded in the 4-ball, with boundary the given knot. We compute the non-orientable 4-genus for all knots with crossing number 8 or 9. As applications we prove a conjecture of Murakami's and Yasuhara's, and give a new lower bound for the slicing number of knot.
We show that if K is any knot whose Ozsvath-Szabo concordance invariant tau(K) is positive, the all-positive Whitehead double of any iterated Bing double of K is topologically but not smoothly slice. We also show that the all-positive Whitehead double of any iterated Bing double of the Hopf link (e.g., the all-positive Whitehead double of the Borromean rings) is not smoothly slice; it is not known whether these links are topologically slice.
A knot in the three-sphere is doubly slice if it is the cross-section of an unknotted two-sphere in the four-sphere. For low-crossing knots, the most complete work to date gives a classification of doubly slice knots through 9 crossings. We extend that work through 12 crossings, resolving all but four cases among the 2,977 prime knots in that range. The techniques involved in this analysis include considerations of the Alexander module and signature functions as well as calculations of the twisted Alexander polynomials for higher-order branched covers. We give explicit illustrations of the dou…
To each rail knotoid we associate two unoriented knots along with their oriented counterparts, thus deriving invariants for rail knotoids based on these associations. We then translate them to invariants of rail isotopy for rail arcs.
We extend the theory of Vassiliev (or finite type) invariants for knots to knotoids using two different approaches. Firstly, we take closures on knotoids to obtain knots and we use the Vassiliev invariants for knots, proving that these are knotoid isotopy invariant. Secondly, we define finite type invariants directly on knotoids, by extending knotoid invariants to singular knotoid invariants via the Vassiliev skein relation. Then, for spherical knotoids we show that there are non-trivial type-1 invariants, in contrast with classical knot theory where type-1 invariants vanish. We give a complet…
We introduce two sequences of two-variable polynomials $\{ L^n_K (t, \ell)\}_{n=1}^{\infty}$ and $\{ F^n_K (t, \ell)\}_{n=1}^{\infty}$, expressed in terms of index value of a crossing and $n$-dwrithe value of a virtual knot $K$, where $t$ and $\ell$ are variables. Basing on the fact that $n$-dwrithe is a flat virtual knot invariant we prove that $L^n_K$ and $F^n_K$ are virtual knot invariants containing Kauffman affine index polynomial as a particular case. Using $L^n_K$ we give sufficient conditions when virtual knot does not admit cosmetic crossing change.
This paper deals with the study of a new family of knot invariants: the $L^2$-Alexander invariant. A main result is to give a method of computation of the $L^2$-Alexander invariant of a knot complement using any presentation of default 1 of the knot group.
A homogeneous knot is a generalization of alternating knots and positive knots. We determine the Rasmussen invariant of a homogeneous knot. This is a new class of knots such that the Rasmussen invariant is explicitly described in terms of its diagrams. As a corollary, we obtain some characterizations of a positive knot. In particular, we recover Baader's theorem which states that a knot is positive if and only if it is homogeneous and strongly quasipositive.
We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) or G2 invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For virtual knots and graphs these invariants provide new graphical information that allows one to prove minimality theorems and to construct new invariants for free knots (unoriented and unlabeled Gauss codes taken up to abstract Reidemeister moves). A novel feature of this approach is that some knots are of sufficient complexity that they evaluate themselves in the…
We define and study Vassiliev invariants for (long) Morse knots. It is shown that there are Vassiliev invariants which can distinguish some topologically equivalent Morse knots. In particular, there is an invariant of order 3 for Morse knots with one maximum that distinguishes two different representations of the figure eight knot. We also present the results of computer calculations for some invariants of low order. It turns out that for Morse knots with two maxima there is a Z/2-valued invariant of order 6 which is not a reduction of any integer-valued invariant.
We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams.
We give a formula of the Upsilon invariant of any L-space cable knot $K_{p,q}$ using $p,Υ_K$ and $Υ_{T_{p,q}}$. The integral value of the Upsilon invariant gives a ${\mathbb Q}$-valued knot concordance invariant. We compute the integral values for L-space iterated cable knots.
It is known that knot Floer homology detects the genus and Alexander polynomial of a knot. We investigate whether knot Floer homology of $K$ detects more structure of minimal genus Seifert surfaces for $K$. We define an invariant of algebraically slice, genus one knots and provide examples to show that knot Floer homology does not detect this invariant. Finally, we remark that certain metabelian $L^2$-signatures bound this invariant from below.
For a knot in the 3-sphere, the Upsilon invariant is a piecewise linear function defined on the interval [0,2]. For an L-space knot, the Upsilon invariant is determined only by the Alexander polynomial of the knot. We exhibit infinitely many pairs of hyperbolic L-space knots such that two knots of each pair have distinct Alexander polynomials, so they are not concordant, but share the same Upsilon invariant. Conversely, we examine the restorability of the Alexander polynomial of an L-space knot from the Upsilon invariant through the Legendre-Fenchel transformation.
This paper introduces two virtual knot theory ``analogues'' of a well-known family of invariants for knots in thickened surfaces: the Grishanov-Vassiliev finite-type invariants of order two. The first, called the three loop isotopy invariant, is an invariant of virtual knots while the second, called the three loop framed isotopy invariant, is a regular isotopy invariant of framed virtual knots. The properties of these invariants are investigated at length. In addition, we make precise the informal notion of ``analogue''. Using this formal definition, it is proved that a generalized three loop …
Using the theory of involutive Heegaard Floer knot theory developed by Hendricks-Manolescu, we define two involutive analogs of the Upsilon knot concordance invariant of Ozsvath-Stipsicz-Szabo. These involutive invariants are piecewise linear functions defined on the interval [0,2]. Each is a concordance invariant and provides bounds on the three-genus of a knot.
We look into computational aspects of two classical knot invariants. We look for ways of simplifying the computation of the coloring invariant and of the Alexander module. We support our ideas with explicit computations on pretzel knots.
The knot invariant Upsilon, defined by Ozsvath, Stipsicz, and Szabo, induces a homomorphism from the smooth knot concordance group to the group of piecewise linear functions on the interval [0,2]. Here we define a set of related secondary invariants, each of which assigns to a knot a piecewise linear function on [0,2]. These secondary invariants provide bounds on the genus and concordance genus of knots. Examples of knots for which Upsilon vanishes but which are detected by these secondary invariants are presented.
In this work we describe a new invariant of virtual knots. We show that this transcendental function invariant generalizes several polynomial invariants of virtual knots, such as the writhe polynomial, the affine index polynomial and the zero polynomial.
We refine the Polyak-Viro Gauss diagram formula for the Vassiliev invariant of order two in a very simple way for the 2-cable of a framed long knot. Surprisingly, the resulting isotopy invariant of framed knots can detect already the non-invertibility of knots. This makes the natural generalization of our invariant for all n-component string link satellites of a framed long knot to a candidate for distinguishing all classical knots.
State-sum invariants for knotted curves and surfaces using quandle cohomology were introduced by Laurel Langford and the authors in math.GT/9903135 In this paper we present methods to compute the invariants and sample computations. Computer calculations of cohomological dimensions for some quandles are presented. For classical knots, Burau representations together with Maple programs are used to evaluate the invariants for knot table. For knotted surfaces in 4-space, movie methods and surface braid theory are used. Relations between the invariants and symmetries of knots are discussed.
We present two different constructions of invariants for Legendrian knots in the standard contact space $\R^3$. These invariants are defined combinatorially, in terms of certain planar projections, and are useful in distinguishing Legendrian knots that have the same classical invariants but are not Legendrian isotopic.
We set forth a definition of hyperfinite knots. Loosely speaking, these are limits of certain sequences of knots with increasing crossing number. These limits exist in appropriate closures of quotient spaces of knots. We give examples of hyperfinite knots. These examples stem from an application of the Thermodynamic Limit to the CJKLS invariant of knots.
We introduce a new algebraic topological technique to detect non-fibred knots in the three sphere using the twisted Alexander invariants. As an application, we show that for any Seifert matrix of a knot with a nontrivial Alexander polynomial, there exist infinitely many non-fibered knots with the given Seifert matrix. We illustrate examples of knots that have trivial Alexander polynomials but do not have twisted Alexander invariants of fibred knots.
Vassiliev introduced filtered invariants of knots using an unknotting operation, called crossing changes. Goussarov, Polyak, and Viro introduced other filtered invariants of virtual knots, which order is called GPV-order, using an unknotting operation, called virtualization. We defined other filtered invariants, which order is called $F$-order, of virtual knots using an unknotting operation, called forbidden moves. In this paper, we show that the set of virtual knot invariants of $F$-order $\le n+1$ is strictly stronger than that of $F$-order $\le n$ and that of GPV-order $\le 2n+1$. To obtain…
I present a formula for the Casson invariant of knots associated with divides. The formula is written in terms of Arnold's invariants of pieces of the divide. Various corollaries are discussed.
In this article, we consider the maximal value of the Thurston--Bennequin invariant of Legendrian knots which topologically represent a fixed knot type in the standard contact 3-space and we prove a formula of the value under the connected sum operation of knots.
We generalize C. Van Cott's results on the $τ$ and $s$-invariants of cabled knots to apply to general satellite knots. This paper uses no Heegaard-Floer homology, relying on more geometric techniques.
We introduce a theory of virtual Legendrian knots. A virtual Legendrian knot is a cooriented wavefront on an oriented surface up to Legendrian isotopy of its lift to the unit cotangent bundle and stabilization and destablization of the surface away from the wavefront. We show that the groups of Vassiliev invariants of virtual Legendrian knots and of virtual framed knots are isomorphic. In particular, Vassiliev invariants cannot be used to distinguish virtual Legendrian knots that are isotopic as virtual framed knots and have equal virtual Maslov numbers.
The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn this can be thought of as a generalization of the quandle 2-cocycle invariant for finite quandles. If the group is a topological group then this invariant can be thought of a topological generalization of the 2-cocycle invariant. The longitudinal mapping invariant is based on a meridian-longitude pair in the knot group. We also give an interpretation of the invariant in terms of quandle colorings of a 1-tangle for gene…
We construct an invariant of virtual knots which is a sliceness obstruction and sensitive to the $Δ$-move. This invariants works if $\Z_{2}\oplus \Z_{2}$-index of chords is present.
For ordinary knots in R3, there are no degree one Vassiliev invariants. For virtual knots, however, the space of degree one Vassiliev invariants is infinite dimensional. We introduce a sequence of three degree one Vassiliev invariants of virtual knots of increasing strength. We demonstrate that the strongest invariant is a universal Vassiliev invariant of degree one for virtual knots in the sense that any other degree one Vassiliev invariant can be recovered from it by a certain natural construction. To prove these results, we extend the based matrix invariant introduced by Turaev for virtual …
We define a filtration on the vector space spanned by Seifert matrices of knots related to Vassiliev's filtration on the space of knots. Further we show that the invariants of knots derived from the filtration can be expressed by coefficients of the Alexander polynomial.
We construct knot invariants from the radical part of projective modules of restricted quantum groups. We also show a relation between these invariants and the colored Alexander invariants.
This short survey, which was written to accompany a minicourse at the BIRS conference "Topology in dimension 4.5", concerns invariants of knotted $2$-spheres in $S^4$, also known as $2$-knots. It covers invariants extracted from the algebraic topology of the knot exterior, including Alexander invariants, the Farber-Levine pairing and Casson-Gordon invariants, as well as gauge theoretic and combinatorial invariants. Details are scarce and new results inexistant.
We relate certain abelian invariants of a knot, namely the Alexander polynomial, the Blanchfield form, and the Arf invariant, to intersection data of a Whitney tower in the 4-ball bounded by the knot. We also give a new 3-dimensional algorithm for computing these invariants.
The homology and cohomology of quandles and racks are used in knot theory: given a finite quandle and a cocycle, we can construct a knot invariant. This is a quick introductory survey to the invariants of knots derived from quandles and racks.
