We consider the space of all smooth knots in the 3-sphere isotopic to a given knot, with the aim of finding a small subspace onto which this large space deformation retracts. For torus knots and many hyperbolic knots we show the subspace can be taken to be the orbit of a single maximally symmetric placement of the knot under the action of SO(4) by rotations of the ambient 3-sphere. This would hold for all hyperbolic knots if it were known that there are no exotic free actions of a finite cyclic group on the 3-sphere. For satellite knots the situation is more complicated but still describable in fairly simple terms. (This preliminary version of the paper does not include details for the case of satellite knots.)
We discuss an "extrinsic" property of knots in a 3-subspace of the 3-sphere $S^3$ to characterize how the subspace is embedded in $S^3$. Specifically, we show that every knot in a subspace of the 3-sphere is transient if and only if the exterior of the subspace is a disjoint union of handlebodies, i.e. regular neighborhoods of embedded graphs, where a knot in a 3-subspace of $S^3$ is said to be transient if it can be moved by a homotopy within the subspace to the trivial knot in $S^3$. To show this, we discuss relation between certain group-theoretic and homotopic properties of knots in a comp…
The inclusion of the space of all knots of a prescribed writhe in a particular isotopy class into the space of all knots in that isotopy class is a weak homotopy equivalence.
The stable 4-genus of a knot K in 3-space is the limiting value of g_4(nK)/n, where g_4 denotes the 4-genus and n goes to infinity. This induces a seminorm on CQ, the concordance group tensored with the rational numbers. Basic properties of the stable genus are developed, as are examples focused on understanding the unit ball under this seminorm for specified subspaces of CQ. Subspaces spanned by torus knots are used to illustrate the distinction between the smooth and topological categories. A final example is given in which Casson-Gordon invariants are used to demonstrate that the stable gen…
For a positive integer $n\ge 3$, the collection of $n$-sided polygons embedded in $3$-space defines the space of geometric knots. We will consider the subspace of equilateral knots, consisting of embedded $n$-sided polygons with unit length edges. Paths in this space determine isotopies of polygons, so path-components correspond to equilateral knot types. When $n\le 5$, the space of equilateral knots is connected. Therefore, we examine the space of equilateral hexagons. Using techniques from symplectic geometry, we can parametrize the space of equilateral hexagons with a set of measure preserv…
Budney recently constructed an operad that encodes splicing of knots. He further showed that the space of (long) knots is generated over this operad by the space of torus knots and hyperbolic knots, thus generalizing the satellite decomposition of knots from isotopy classes to the level of the space of knots. Infection by string links is a generalization of splicing from knots to links. We construct a colored operad that encodes string link infection. We prove that a certain subspace of the space of 2-component string links is generated over a suboperad of our operad by its subspace of prime l…
We study two sorts of actions on the space of conjugacy classes of irreducible $SU_2$-representations of a knot group. One of them is an involution which comes from the algebraic structure of $SU_2$ and the other is the action by the outer automorphism group of the knot group. In particular, we consider them on an 1-dimensional smooth part of the space, which is canonically oriented and metrized via a Reidemeister torsion volume form. As an application we show that the Reidemeister torsion function on the 1-dimensional subspace has symmetry about the metrization.
We study the alternating subspace of holomorphic sections of a special prequantum line bundle over SU(2)-character variety of torus, and show that it is isomorphic to the projective representation of mapping class group of peripheral torus given by the SO(3) Witten-Chern-Simons theory. We conjecture that the large r asymptotics of $L^2$-norm of SO(3)-knot states via geometric quantization capture the simplicial volume of knot complements.
In this short note we use methods of Friedl, Livingston and Zentner to show that there are knots that are not algebraically concordant to a connected sum of positive and negative L-space knots.
Algebraic knots are known to be iterated torus knots and to admit L-space surgeries. However, Hedden proved that there are iterated torus knots that admit L-space surgeries but are not algebraic. We present an infinite family of such examples, with the additional property that no nontrivial linear combination of knots in this family is concordant to a linear combination of algebraic knots. The proof uses the Ozsvath-Stipsicz-Szabo Upsilon function, and also introduces a new invariant of L-space knots, the formal semigroup.
It is known that connected sums of positive torus knots are not concordant to $L$-space knots. Here we consider differences of torus knots. The main result states that the subgroup of the concordance group generated by two positive torus knots contains no nontrivial $L$-space knots other than the torus knots themselves. Generalizations to subgroups generated by more than two torus knots are also considered.
A knot in the 3-sphere is called an L--space knot if it admits a nontrivial Dehn surgery yielding an L--space. Like torus knots and Berge knots, many L--space knots admit also a Seifert fibered surgery. We give a concrete example of a hyperbolic, L-space knot which has no exceptional surgeries, in particular, no Seifert fibered surgeries.
Divide knots and links, defined by A'Campo in the singularity theory of complex curves, is a method to present knots or links by real plane curves. The present paper is a continuation of the author's previous result that every knot in the major subfamilies of Berge's lens space surgery (i.e., knots yielding a lens space by Dehn surgery) is presented by an L-shaped curve as a divide knot. In the present paper, L-shaped curves are generalized and it is shown that every knot in the minor subfamilies, called sporadic examples, of Berge's lens space surgery is presented by a generalized L-shaped cu…
Using Hirasawa-Murasugi's classification of fibered Montesinos knots we classify the L-space Montesinos knots, providing further evidence towards a conjecture of Lidman-Moore that L-space knots have no essential Conway spheres. In the process, we classify the fibered Montesinos knots whose open books support the tight contact structure on $S^3$. We also construct L-space knots with arbitrarily large tunnel number and discuss the question of whether L-space knots admit essential tangle decompositions in the context of satellite operations and tunnel number.
Let $K$ be a hyperbolic knot in the 3-sphere. If $r$-surgery on $K$ yields a lens space, then we show that the order of the fundamental group of the lens space is at most $12g-7$, where $g$ is the genus of $K$. If we specialize to genus one case, it will be proved that no lens space can be obtained from genus one, hyperbolic knots by Dehn surgery. Therefore, together with known facts, we have that a genus one knot $K$ admits Dehn surgery yielding a lens space if and only if $K$ is the trefoil.
We give sufficient conditions for a satellite knot to admit an L-space surgery, and use this result to give new infinite families of patterns which produce satellite L-space knots.
For any alternating knot, it is known that the double branched cover of the $3$-sphere branched over the knot is an $L$-space. We show that the three-fold cyclic branched cover is also an $L$-space for any genus one alternating knot.
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.
In 2008, Kauffman and Lomonaco introduce the concepts of a knot mosaic and the mosaic number of a knot or link, the smallest integer $n$ such that a knot or link can be represented on an $n$-mosaic. In arXiv:1702.06462, the authors explore space-efficient knot mosaics and the tile number of a knot or link, the smallest number of non-blank tiles necessary to depict the knot or link on a mosaic. They determine bounds for the tile number in terms of the mosaic number. In this paper, we focus specifically on prime knots with mosaic number 6. We determine a complete list of these knots, provide a m…
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.
We show that there exist infinitely many pairs of distinct knots in the 3-sphere such that each pair can yield homeomorphic lens spaces by the same Dehn surgery. Moreover, each knot of the pair can be chosen to be a torus knot, a satellite knot or a hyperbolic knot, except that both cannot be satellite knots simultaneously. This exception is shown to be unavoidable by the classical theory of binary quadratic forms.
The concepts of tile number and space-efficiency for knot mosaics were first explored by Heap and Knowles (arXiv:1702.06462), where they determined the possible tile numbers and space-efficient layouts for every prime knot with mosaic number 6 or less. In this paper, we extend those results to prime knots with mosaic number 7.
Let $P(K)$ be a satellite knot where the pattern, $P$, is a Berge-Gabai knot (i.e., a knot in the solid torus with a non-trivial solid torus Dehn surgery), and the companion, $K$, is a non-trivial knot in $S^3$. We prove that $P(K)$ is an L-space knot if and only if $K$ is an L-space knot and $P$ is sufficiently positively twisted relative to the genus of $K$. This generalizes the result for cables due to Hedden and the first author.
We determine the non-null homologous knots in lens spaces whose exteriors contain properly embedded once-punctured tori. All such knots arise as surgeries on the Whitehead link and are grid number 1 in their lens spaces. As a corollary, we classify once-punctured torus bundles that admit a lens space filling.
Greene, Lewallen and Vafaee characterized $(1,1)$ L-space knots in $S^3$ and lens space in the notation of coherent reduced $(1,1)$-diagrams. We analyze these diagrams, and deduce an explicit description of these knots. With the new description, we prove that any L-space obtained by Dehn surgery on a $(1,1)$-knot in $S^3$ has non-left-orderable fundamental group.
For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, is an embedded torus. Given a thick configuration K, perturbations of size r<R(K) define satellite structures, or local knotting. We explore knotting within these tubes both theoretically and numerically. We provide bounds on perturbation radii for which we can see small trefoil and figure-eight summands and use Monte Carlo simulations to approximate the relative probabilities of these structures as a function of the number of edges.
We propose a classification of knots in S^1 x S^2 that admit a longitudinal surgery to a lens space. Any lens space obtainable by longitudinal surgery on some knots in S^1 x S^2 may be obtained from a Berge-Gabai knot in a Heegaard solid torus of S^1 x S^2, as observed by Rasmussen. We show that there are yet two other families of knots: those that lie on the fiber of a genus one fibered knot and the `sporadic' knots. All these knots in S^1 x S^2 are both doubly primitive and spherical braids. This classification arose from generalizing Berge's list of doubly primitive knots in S^3, though we …
Let $D$ be a diagram of an alternating knot with unknotting number one. The branched double cover of $S^3$ branched over $D$ is an L-space obtained by half integral surgery on a knot $K_D$. We denote the set of all such knots $K_D$ by $\mathcal D$. We characterize when $K_D\in \mathcal D$ is a torus knot, a satellite knot or a hyperbolic knot. In a different direction, we show that for a given $n>0$, there are only finitely many L-space knots in $\mathcal D$ with genus less than $n$.
In this paper we introduce the concept of a space-efficient knot mosaic. That is, we seek to determine how to create knot mosaics using the least number of non-blank tiles necessary to depict the knot. This least number is called the tile number of the knot. We determine strict bounds for the tile number of a knot in terms of the mosaic number of the knot. In particular, if $t$ is the tile number of a prime knot with mosaic number $m$, then $5m-8 \leq t \leq m^2-4$ if $m$ is even and $5m-8 \leq t \leq m^2-8$ if $m$ is odd. We also determine the tile number of several knots and provide space-ef…
Ozsváth-Szabó proved the property that any coefficient of Alexander polynomial of lens space knot is either $\pm1$ or $0$ and the non-zero coefficients are alternating. Combining the formulas of the Alexander polynomial of lens space knots due to Kadokami-Yamada and Ichihara-Saito-Teragaito, we refine Ozsváth-Szabó's property as the existence of simple curves included in a region in ${\Bbb R}^2$. The existence of curves, that has no end-points connected, is just 1-component in a region, can search distribution of non-zero coefficients of the Alexander polynomial of the lens space knot. This cu…
Using computational techniques we tabulate prime knots up to five crossings in the solid torus and the infinite family of lens spaces $L(p,q)$. For these knots we calculate the second and third skein module and establish which prime knots in the solid torus are amphichiral. Most knots are distinguished by the skein modules. For the handful of cases where the skein modules fail to detect inequivalent knots, we calculate and compare the hyperbolic structures of the knot complements. We were unable to resolve a handful of 5-crossing cases for $p\geq 13$.
We develop homological techniques for finding explicit combinatorial expressions of finite-type cohomology classes of spaces of knots in $R^n, n \ge 3,$ generalizing Polyak--Viro formulas for invariants (i.e. 0-dimensional cohomology classes) of knots in $R^3$. As the first applications we give such formulas for the (reduced mod 2) {\em generalized Teiblum--Turchin cocycle} of order 3 (which is the simplest cohomology class of {\em long knots} $R^1 \hookrightarrow R^n$ not reducible to knot invariants or their natural stabilizations), and for all integral cohomology classes of orders 1 and 2 o…
We construct the first examples of asymmetric L-space knots in $S^3$. More specifically, we exhibit a construction of hyperbolic knots in $S^3$ with both (i) a surgery that may be realized as a surgery on a strongly invertible link such that the result of the surgery is the double branched cover of an alternating link and (ii) trivial isometry group. In particular, this produces L-space knots in $S^3$ which are not strongly invertible. The construction also immediately extends to produce asymmetric L-space knots in any lens space, including $S^1 \times S^2$.
Given an L-space knot we show that its Upsilon function is the Legendre transform of a counting function equivalent to the d-invariants of its large surgeries. The unknotting obstruction obtained for the Upsilon function is, in the case of L-space knots, contained in the d-invariants of large surgeries. Generalizations apply for connected sums of L-space knots, which imply that the slice obstruction provided by Upsilon on the subgroup of concordance generated by L-space knots is no finer than that provided by the d-invariants.
We consider the following question: when is the manifold obtained by gluing together two knot complements an $L$-space? Hedden and Levine proved that splicing 0-framed complements of nontrivial knots never produces an $L$-space. We extend this result to allow for arbitrary integer framings. We find that splicing two integer framed nontrivial knot complements only produces an $L$-space if both knots are $L$-space knots and the framings lie in an appropriate range. The proof involves a careful analysis of the bordered Heegaard Floer invariants of each knot complement.
We show that the knot lattice homology of a knot in an L-space is equivalent to the knot Floer homology of the same knot (viewed these invariants as filtered chain complexes over the polynomial ring Z/2Z [U]). Suppose that G is a negative definite plumbing tree which contains a vertex w such that G-w is a union of rational graphs. Using the identification of knot homologies we show that for such graphs the lattice homology HF(G)$ is isomorphic to the Heegaard Floer homology HF^-(Y_G) of the corresponding rational homology sphere Y_G.
There is a higher dimensional analogue of the perturbative Chern-Simons theory in the sense that a similar perturbative series as in 3-dimension, which is computed via configuration space integral, yields an invariant of higher dimensional knots (Bott-Cattaneo-Rossi invariant), which is constructed by Bott for degree 2 and by Cattaneo-Rossi for higher degrees. However, its feature is yet unknown. In this paper we restrict the study to long ribbon n-knots and characterize the Bott-Cattaneo-Rossi invariant as a finite type invariant of long ribbon n-knots in [HKS]. As a consequence, we obtain a …
Using an invariant defined by Rasmussen, we extend an argument given by Hedden and Watson which further restricts which Alexander polynomials can be realized by $L$-space knots.
It is proved that every knot in the major subfamilies of J. Berge's lens space surgery (i.e., knots yielding a lens space by Dehn surgery) is presented by an L-shaped (real) plane curve as a "divide knot" defined by N. A'Campo in the context of singularity theory of complex curves. For each knot given by Berge's parameters, the corresponding plane curve is constructed. The surgery coefficients are also considered. Such presentations support us to study each knot itself, and the relationship among the knots in the set of lens space surgeries.
A rational homology sphere whose Heegaard Floer homology is the same as that of a lens space is called an L-space. We classify pretzel knots with any number of tangles which admit L-space surgeries. This rests on Gabai's classification of fibered pretzel links.
This article surveys the use of configuration space integrals in the study of the topology of knot and link spaces. The main focus is the exposition of how these integrals produce finite type invariants of classical knots and links. More generally, we also explain the construction of a chain map, given by configuration space integrals, between a certain diagram complex and the deRham complex of the space of knots in dimension four or more. A generalization to spaces of links, homotopy links, and braids is also treated, as are connections to Milnor invariants, manifold calculus of functors, and…
Hedden defined two knots in each lens space that, through analogies with their knot Floer homology and doubly pointed Heegaard diagrams of genus one, may be viewed as generalizations of the two trefoils in S^3. Rasmussen shows that when the `left-handed' one is in the homology class of the dual to a Berge knot of type VII, it admits an L-space homology sphere surgery. In this note we give a simple proof that these L-space homology spheres are always the Poincaré homology sphere.
In this survey we summarize results regarding the Kauffman bracket, HOMFLYPT, Kauffman 2-variable and Dubrovnik skein modules, and the Alexander polynomial of links in lens spaces, which we represent as mixed link diagrams. These invariants generalize the corresponding knot polynomials in the classical case. We compare the invariants by means of the ability to distinguish between some difficult cases of knots with certain symmetries.
In this paper, we consider the knot complement problem for not null-homologous knots in homology lens spaces. Let $M$ be a homology lens space with $H_1(M; \mathbb{Z}) \cong \mathbb{Z}_p$ and $K$ a not null-homologous knot in $M$. We show that $K$ is determined by its complement if $M$ is non-hyperbolic, $K$ is hyperbolic, and $p$ is a prime more than 7, or, if $M$ is actually a lens space $L(p,q)$ and $K$ represents a generator of $H_1(L(p,q))$.
Closed 3-string braids admit many bandings to two-bridge links. By way of the Montesinos Trick, this allows us to construct infinite families of knots in the connected sum of lens spaces L(r,1) # L(s,1) that admit a surgery to a lens space for all pairs of integers (r,s) except (0,0). These knots are typically hyperbolic. We also demonstrate that the previously known two families of examples of hyperbolic knots in non-prime manifolds with lens space surgeries of Eudave-Munoz--Wu and Kang all fit this construction. As such, we propose a generalization of the Cabling Conjecture of Gonzales-Acuna…
We prove that there are exactly $6$ Nil Seifert fibred spaces which can be obtained by Dehn surgeries on non-trefoil knots in $S^3$, with $\{60, 144, 156, 288, 300\}$ as the exact set of all such surgery slopes up to taking the mirror images of the knots. We conjecture that there are exactly $4$ specific hyperbolic knots in $S^3$ which admit Nil Seifert fibred surgery. We also give some more general results and a more general conjecture concerning Seifert fibred surgeries on hyperbolic knots in $S^3$.
In the classical knot theory there is a well-known notion of descending diagram. From an arbitrary diagram one can easily obtain, by some crossing changes, a descending diagram which is a diagram of the unknot or unlink. In this paper the notion of descending diagram for knots and links in the real space is extended to the case of nonoriented knots and links in the projective space. It is also shown that this notion cannot be extended to oriented links.
