26 pages, 10 figures. Revision 7 Sep 99: added discussion on Whitehead doubles, the Casson invariant and signature, and some inequalitites related to the genus, crossing, and unknotting number; revision 1 Mar 01: added application to Legendrian knots and property P conjecture
Using the Fiedler-Polyak-Viro Gauss diagram formulas we study the Vassiliev invariants of degree 2 and 3 on almost positive knots. As a consequence we show that the number of almost positive knots of given genus or unknotting number grows polynomially in the crossing number, and also recover and extend, inter alia to their untwisted Whitehead doubles, previous results on the polynomials and signatures of such knots. In particular, we prove that there are no achiral almost positive knots and classify all almost positive diagrams of the unknot. We give an application to contact geometry (Legendrian knots) and property P.
We consider the question, asked by Friedl, Livingston and Zentner, of which sums of torus knots are concordant to alternating knots. After a brief analysis of the problem in its full generality, we focus on sums of two torus knots. We describe some effective obstructions based on Heegaard Floer homology.
Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in $S^3$ do not admit purely cosmetic surgeries. In this article, we confirm this conjecture for connected sums of knots by analysing the JSJ-structures.
We prove a formula for the conjugation action on the knot Floer complex of the connected sum of two knots. Using the formula we construct a homomorphism from the smooth concordance group to an abelian group consisting of chain complexes with homotopy automorphisms, modulo an equivalence relation. Using our connected sum formula, we perform some example computations of Hendricks and Manolescu's involutive invariants on large surgeries of connected sums of knots.
We give a short proof that if a non-trivial band sum of two knots results in a tight fibered knot, then the band sum is a connected sum. In particular, this means that any prime knot obtained by a non-trivial band sum is not tight fibered. Since a positive L-space knot is tight fibered, a non-trivial band sum never yields an L-space knot. Consequently, any knot obtained by a non-trivial band sum cannot admit a finite surgery. For context, we exhibit two examples of non-trivial band sums of tight fibered knots producing prime knots: one is fibered but not tight, and the other is strongly quasip…
For a knot $K,$ a slope $r$ is said to be characterizing if for no other knot $J$ does $r$-framed surgery along $J$ yield the same manifold as $r$-framed surgery on $K.$ Applying a condition of Baker and Motegi, we show that the knots $T_{2,2n+3}\#T_{-2,2n+1}$ have infinitely many non-characterizing slopes.
This paper studies the question of whether minimal genus Heegaard splittings of exterior spaces of knots which are connected sums are weakly reducible or not. Furthermore it is shown that the Heegaard splittings of the knots used by Morimoto to show that tunnel number can be sub-additive are all strongly irreducible. These are the first examples of strongly irreducible minimal genus Heegaard splittings of composite knots. We also give a characterization of when is a set of primitive annuli on a handlebody simultaneously primitive. This characterization is different from that given in [Go].
In this paper, we generalize a result of Satoh to show that for any odd natural $n$, the connected sum of the $n$-twist spun sphere of a knot $K$ and an unknotted projective plane in the 4-sphere is equivalent to the same unknotted projective plane. We additionally provide a fix to a small error in Satoh's proof of the case that $K$ is a 2-bridge knot.
A fixed knot $K$ acts via Murasugi sum on the space $\mathcal{S}$ of isotopy classes of knots. This operation endows $\mathcal{S}$ with a directed graph structure denoted by $M\kern-1pt SG(K)$. We show that any given family of knots in $M\kern-1pt SG(K)$ has the structure of a bi-directed complete graph, which is not the case if we restrict the complexity of Murasugi sums. For that purpose, we show that any knot is a Murasugi sum of any two knots, and we give lower and upper bounds for the minimal complexity of Murasugi sum to obtain $K_3$ by $K_1$ and $K_2$. As an application, we show that gi…
Murasugi sums can be defined as readily for Morse maps to the circle of (arbitrary) link complements in the 3-sphere as for fibrations over the circle of (fibered) link complements in the 3-sphere. As one application, I show that if a knot K has free genus m, then there is a Morse map from its complement to the circle (representing the relative homology class of a Seifert surface for K) with no more than 4m critical points.
We study the behavior of Legendrian and transverse knots under the operation of connected sums. As a consequence we show that there exist Legendrian knots that are not distinguished by any known invariant. Moreover, we classify Legendrian knots in some non-Legendrian simple knot types.
We study the four-genus of linear combinations of torus knots: aT(p,q) # -bT(p',q'). Fixing positive p, q, p', and q', our focus is on the behavior of the four-genus as a function of positive a and b. Three types of examples are presented: in the first, for all a and b the four-genus is completely determined by the Tristram-Levine signature function; for the second, the recently defined Upsilon function of Ozsvath-Stipsicz-Szabo determines the four-genus for all a and b; for the third, a surprising interplay between signatures and Upsilon appears.
We consider the recently introduced knotting-unknotting game, in which two players take turns resolving crossings in a knot diagram which initially is missing all its crossing information. Once the knot is fully resolved, the winner is decided by whether the knot is equivalent to the unknot. In this paper we determine which player wins under perfect play in the initial positions (knot shadows) that are guaranteed to produce rational knots or sums of rational knots. This is accomplished in part by an analysis of the effects of pseudo Reidemeister moves on positions, and a computer verification …
We show that the SU(3) Casson invariant for spliced sums along certain torus knots equals 16 times the product of their SU(2) Casson knot invariants. The key step is a splitting formula for su(n) spectral flow for closed 3-manifolds split along a torus.
We define a "reduced" version of the knot Floer complex $CFK^-(K)$, and show that it behaves well under connected sums and retains enough information to compute Heegaard Floer $d$-invariants of manifolds arising as surgeries on the knot $K$. As an application to connected sums, we prove that if a knot in the three-sphere admits an $L$-space surgery, it must be a prime knot. As an application of the computation of $d$-invariants, we show that the Alexander polynomial is a concordance invariant within the class of $L$-space knots, and show the four-genus bound given by the $d$-invariant of +1-su…
Given $(V_1,V_2)$ a Heegaard splitting of the complement of a composite knot $K=K_1# K_2$ in $S^3$, where $K_i, i=1,2$ are prime knots, we have a unique, up to isotopy, decomposing annulus $A$. When the intersection of $A$ and $V_1$ is a minimal collection of disks we study the components of $V_1-N(A)$ and show that at most one component is a 3-ball meeting $A$ in two disks. This is a crucial step in proving the conjecture that a necessary and sufficient condition for the tunnel number of a connected sum to be less than or equal to the sum of the tunnel numbers is that one of the knots has a H…
We show that the triple-crossing number of any knot is greater or equal to twice its (canonical) genus and we show an even stronger bound in the case of links. As an application we show that this bound is strong enough to obtain the triple-crossing numbers of all torus knots, and of many more knots and their connected sums.
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.
We show that there exist infinitely many examples of pairs of knots, K_1 and K_2, that have no epimorphism $π_1(S^3\setminus K_1) \to π_1(S^3\setminus K_2)$ preserving peripheral structure although their A-polynomials have the factorization $A_{K_2}(L,M) \mid A_{K_1}(L,M)$. Our construction accounts for most of the known factorizations of this form for knots with 10 or fewer crossings. In particular, we conclude that while an epimorphism will lead to a factorization of A-polynomials, the converse generally fails.
We establish a formula for the SL(2,C) Casson invariant of spliced sums of homology spheres along knots. Along the way, we show that the SL(2,C) Casson invariant vanishes for spliced sums along knots in the 3-sphere.
We define a new kind of Gauss diagrams to describe knots in the solid torus with projections in the annulus. We see that it provides an efficient tool for showing that a knot diagram can be fully recovered from its decorated Gauss diagram, and we use it to establish a characterization of the decorated Gauss diagrams of closed braids.
We show that if a fibered knot $K$ is expressed as a band--connected sum of $K_1, \ldots, K_n$, then each $K_i$ is fibered, and the genus of $K$ is greater than or equal to that of the connected sum of $K_1,\ldots,K_n$.
Twisted knot theory, introduced by M.O.Bourgoin, is a generalization of virtual knot theory. It is easily shown that any virtual knot can be deformed into a trivial knot by a finite sequence of generalized Reidemeister moves and two "forbidden moves" $F_{1}$ and $F_{2}$. Similarly, we show that any twisted knot also can be deformed into a trivial knot or a trivial knot with a bar by a finite sequence of extended Reidemeister moves and three "forbidden moves" $T_{4}$, $F_{1}$ (or $F_{2}$) and $F_{3}$ (or $F_{4}$) .
The forbidden moves can be combined with Gauss diagram Reidemeister moves to obtain move sequences with which we may change any Gauss diagram (and hence any virtual knot) into any other, including in particular the unknotted diagram
We give a formula for the radial asymptotics to all orders of the special $q$-hypergeometric series known as Nahm sums at complex roots of unity. This result is used in~\cite{CGZ} to prove one direction of Nahm's conjecture relating the modularity of Nahm sums to the vanishing of a certain invariant in $K$-theory. The power series occurring in our asymptotic formula are identical to the conjectured asymptotics of the Kashaev invariant of a knot once we convert Neumann-Zagier data into Nahm data, suggesting a deep connection between asymptotics of quantum knot invariants and asymptotics of Nahm…
We define two new families of invariants for (3-manifold, graph) pairs which detect the unknot and are additive under connected sum of pairs and (-1/2)-additive under trivalent vertex sum of pairs. The first of these families is closely related to both bridge number and tunnel number. The second of these families is a variation and generalization of Gabai's width for knots in the 3-sphere. We give applications to the tunnel number and higher genus bridge number of connected sums of knots.
The theory of Gauss diagrams and Gauss diagram formulas provides convenient ways to compute knot invariants, such as coefficients of the HOMFLYPT polynomial. In \cite{4,5}, the author uses Gauss diagram formulas to find combinatorial 1-cocycles in the moduli space of knots in the solid torus. Evaluated on canonical loops, one can then obtain new, non trivial knot invariants. In those books, the author conjectures that a new formula, based on the Vassiliev invariant $v_3$ also gives a 1-cocycle. We prove that it is in fact true by using the same methods developed by the author in those books.
Given a group endowed with a Z/2-valued morphism we associate a Gauss diagram theory, and show that for a particular choice of the group these diagrams encode faithfully virtual knots on a given arbitrary surface. This theory contains all of the earlier attempts to decorate Gauss diagrams, in a way that is made precise via symmetry-preserving maps. These maps become crucial when one makes use of decorated Gauss diagrams to describe finite-type invariants. In particular they allow us to generalize Grishanov-Vassiliev's formulas and to show that they define invariants of virtual knots.
We investigate cobordisms of free knots. Free knots and links are also called homotopy classes of Gauss words and phrases. We define a new strong invariant of free knots which allows to detect free knots not cobordant to the trivial one.
Recent advances in Quantum Topology assign $q$-series to knots in at least three different ways. The $q$-series are given by generalized Nahm sums (i.e., special $q$-hypergeometric sums) and have unknown modular and asymptotic properties. We give an efficient method to compute those $q$-series that come from planar graphs (i.e., reduced Tait graphs of alternating links) and compute several terms of those series for all graphs with at most 8 edges drawing several conclusions. In addition, we give a graph-theory proof of a theorem of Dasbach-Lin which identifies the coefficient of $q^k$ in those…
We give a number theoretic proof of the integrality of certain BPS invariants of knots. The formulas for these numbers are sums involving binomial coefficients and the Möbius function. We also prove a conjecture about further divisibility properties of the invariants.
Goussarov, Polyak, and Viro proved that finite type invariants of knots are ``finitely multi-local'', meaning that on a knot diagram, sums of quantities, defined by local information, determine the value of the knot invariant. The result implies the existence of Gauss diagram combinatorial formulas for finite type invariants. This article presents a simplified account of the original approach. The simplifications provide an easy generalization to the cases of pure tangles and pure braids. The associated problem on group algebras is introduced and used to prove the existence of ``multi-local wo…
The Slope Conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjecture proposed by Kalfagianni and Tran asserts that the linear term in the degree determines the topology of an essential surface that satisfies the Slope Conjecture. Under certain hypotheses, we show that Mazur doubles of knots satisfy the Strong Slope Conjecture if the original knot does. Consequently, any knot obtained by a finite sequence of cabling, untwisted w--generalized Whitehead doublings with w > 0, connected sum…
Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in $\mathbb{R}^4$; including virtual crossings allows for a theory of virtual knotted surfaces and virtual cobordisms. Biquandle counting invariants are defined only for marked vertex diagrams representing knotted orientable surfaces; we extend these invariants to all virtual marked vertex diagrams by considering colorings by involutory biquandles, also known as bikei. We introduce a way of representing marked vertex diagrams with Gauss diagrams and use these to characterize orientability.
Virtual knot theory is a generalization (discovered by the author in 1996) of knot theory to the study of all oriented Gauss codes. (Classical knot theory is a study of planar Gauss codes.) Graph theory studies non-planar graphs via graphical diagrams with virtual crossings. Virtual knot theory studies non-planar Gauss codes via knot diagrams with virtual crossings. This paper gives basic results and examples (such as non-trivial virtual knots with trivial Jones polynomial), studies fundamental group and quandles of virtual knots, extensions of the bracket and Jones polynomials, quantum link i…
We observe that any knot invariant extends to virtual knots. The isotopy classification problem for virtual knots is reduced to an algebraic problem formulated in terms of an algebra of arrow diagrams. We introduce a new notion of finite type invariant and show that the restriction of any such invariant of degree n to classical knots is an invariant of degree at most n in the classical sense. A universal invariant of degree at most n is defined via a Gauss diagram formula. This machinery is used to obtain explicit formulas for invariants of low degrees. The same technique is also used to prove…
We prove the following results (1) (2) (3) on relations between $n$-links and their components. (1) Let L=(L_1, L_2) be a (4k+1)-link (4k+1\geq 5). Then we have Arf L=Arf L_1+Arf L_2. (2) Let L=(L_1, L_2) be a (4k+3)-link (4k+3\geq3). Then we have σL=σL_1+σL_2. (3) Let n\geq1. Then there is a nonribbon n-link L=(L_1, L_2) such that L_i is a trivial knot. We prove the following results (4) (5) (6) (7) on band-sums of n-links. (4) Let L=(L_1, L_2) be a (4k+1)-link (4k+1\geq 5). Let K be a band-sum of L. Then we have Arf K=Arf L_1+Arf L_2. (5) Let L=(L_1, L_2) be a (4k+3)-link (4k+3\geq3). Let K …
S. Gukov and C. Manolescu conjectured that the Melvin-Morton-Rozansky expansion of the colored Jones polynomials can be re-summed into a two-variable series $F_K(x,q)$, which is the knot complement version of the 3-manifold invariant $\hat{Z}$ whose existence was predicted earlier by S. Gukov, P. Putrov and C. Vafa. In this paper we use an inverted version of the R-matrix state sum to prove this conjecture for a big class of links that includes all homogeneous braid links as well as all fibered knots up to 10 crossings. We also study an inverted version of Habiro's cyclotomic series that leads…
Proteins are linear molecular chains that often fold to function. The topology of folding is widely believed to define its properties and function, and knot theory has been applied to study protein structure and its implications. More that 97% of proteins are, however, classified as unknots when intra-chain interactions are ignored. This raises the question as to whether knot theory can be extended to include intra-chain interactions and thus be able to categorize topology of the proteins that are otherwise classified as unknotted. Here, we develop knot theory for folded linear molecular chain…
