1. Knots and their relatives 2. Knot invariants 3. Finite type invariants 4. Chord diagrams 5. Jacobi diagrams 6. Lie algebra weight systems 7. Algebra of 3-graphs 8. The Kontsevich integral 9. Framed knots and cabling operations 10. The Drinfeld associator 11. The Kontsevich integral: advanced features 12. Braids and string links 13. Gauss diagrams 14. Miscellany 15. The space of all knots Appendix References Notations Index.

/pdf/introduction-to-vassiliev-knot-invariants-3xrn99sd3q.pdf

Introduction to Vassiliev Knot Invariants

The classical abelian invariants of a knot are the Alexander module, which is the first homology group of the the unique infinite cyclic covering space of S 3 - K, considered as a module over the (commutative) Laurent polynomial ring, and the Blanchfield linking pairing defined on this module. From the perspective of the knot group, G, these invariants reflect the structure of G ( 1 ) /G ( 2 ) as a module over G/G ( 1 ) (here G ( n ) is the n t h term of the derived series of G). Hence any phenomenon associated to G ( 2 ) is invisible to abelian invariants. This paper begins the systematic study of invariants associated to solvable covering spaces of knot exteriors, in particular the study of what we call the n t h higher-order Alexander module, G ( + 1 ) /G ( n + 2 ) , considered as a Z[G/G ( n + 1 ) ]-module. We show that these modules share almost all of the properties of the classical Alexander module. They are torsion modules with higher-order Alexander polynomials whose degrees give lower bounds for the knot genus. The modules have presentation matrices derived either from a group presentation or from a Seifert surface. They admit higher-order linking forms exhibiting self-duality. There are applications to estimating knot genus and to detecting fibered, prime and alternating knots. There are also surprising applications to detecting symplectic structures on 4-manifolds. These modules are similar to but different from those considered by the author, Kent Orr and Peter Teichner and are special cases of the modules considered subsequently by Shelly Harvey for arbitrary 3-manifolds.

/pdf/noncommutative-knot-theory-keym3wu0sa.pdf

Noncommutative knot theory.

In this paper we study the knot Floer homology invariants of the twisted and untwisted Whitehead doubles of an arbitrary knot, K . A formula is presented for the filtered chain homotopy type of b HFK.D .K;t// in terms of the invariants for K , where D .K;t/ denotes the t ‐twisted positive (resp. negative)-clasped Whitehead double of K . In particular, the formula can be used iteratively and can be used to compute the Floer homology of manifolds obtained by surgery on Whitehead doubles. An immediate corollary is that . DC.K;t//D1 if t 0 are not smoothly slice. Another corollary is a closed formula for the Floer homology of the three-manifold obtained by gluing the complement of an arbitrary knot, K , to the complement of the trefoil.

/pdf/knot-floer-homology-of-whitehead-doubles-21j3v65z11.pdf

Knot Floer homology of Whitehead doubles

We continue our study of the knot Floer homology invariants of cable knots. For large | n|, we prove that many of the filtered subcomplexes in the knot Floer homology filtration associated to the (p, pn+1) cable of a knot, K, are isomorphic to those of K. This result allows us to obtain information about the behavior of the Ozsvath-Szabo concordance invariant under cabling, which has geometric consequences for the cabling operation. Applications considered include quasipositivity in the braid group, the knot theory of complex curves, smooth concordance, and L-space surgeries.

On Knot Floer Homology and Cabling: 2

We propose and analyze a structure with which to organize the difference between a knot in S 3 bounding a topologically embedded 2‐disk in B 4 and it bounding a smoothly embedded disk. The n‐solvable filtration of the topological knot concordance group, due to Cochran‐Orr‐Teichner, may be complete in the sense that any knot in the intersection of its terms may well be topologically slice. However, the natural extension of this filtration to what is called the n‐solvable filtration of the smooth knot concordance group, is unsatisfactory because any topologically slice knot lies in every term of the filtration. To ameliorate this we investigate a new filtration,fBng, that is simultaneously a refinement of the n‐solvable filtration and a generalization of notions of positivity studied by Gompf and Cochran. We show that each Bn=BnC1 has infinite rank. But our primary interest is in the induced filtration,fTng, on the subgroup, T , of knots that are topologically slice. We prove that T=T0 is large, detected by gauge-theoretic invariants and the , s , ‐invariants, while the nontriviality of T0=T1 can be detected by certain d ‐invariants. All of these concordance obstructions vanish for knots in T1 . Nonetheless, going beyond this, our main result is that T1=T2 has positive rank. Moreover under a “weak homotopy-ribbon” condition, we show that each Tn=TnC1 has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity. 57M25

https://projecteuclid.org/download/pdf_1/euclid.gt/1513732648

Filtering smooth concordance classes of topologically slice knots

Let v be any integer-valued additive knot invariant that bounds the smooth 4-genus of a knot K, |ν(K)| ≤ g 4 (K), and determines the 4-ball genus of positive torus knots, ν(T p,q ) = (p - 1)(q - 1)/2. Either of the knot concordance invariants of Ozsvath-Szabo or Rasmussen, suitably normalized, have these properties. Let D±(K, t) denote the positive or negative t-twisted double of K. We prove that if ν(D + (K,t)) =±1, then ν(D-(K,t)) = 0. It is also shown that ν(D+(K, t)) = 1 for all t ≤ TB(K) and ν(D + (K,t)) = 0 for all t ≥ -TB(-K), where TB(K) denotes the Thurston-Bennequin number. A realization result is also presented: for any 2g x 2g Seifert matrix A and integer a, |a| ≤ g, there is a knot with Seifert form A and v(K) = a.

/pdf/ozsvath-szabo-and-rasmussen-invariants-of-doubled-knots-5fmefevtxf.pdf

Ozsváth-Szabó and Rasmussen invariants of doubled knots

Murasugi discovered two criteria that must be satisfied by the Alexander polynomial of a periodic knot. We generalize these to the case of twisted Alexander polynomials. Examples demonstrate the application of these new criteria, including to knots with trivial Alexander polynomial, such as the two polynomial 1 knots with 11 crossings. Hartley found a restrictive condition satisfied by the Alexander polynomial of any freely periodic knot. We generalize this result to the twisted Alexander polynomial and illustrate the applicability of this extension in cases in which Hartley's criterion does not apply.

/pdf/twisted-alexander-polynomials-of-periodic-knots-1j0lrdij5j.pdf

Twisted Alexander polynomials of periodic knots

This paper gives an algebraic characterization of Alexander polynomials of equivariant ribbon knots and a factorization condition satisfied by Alexander polynomials of equivariant slice knots.

/pdf/alexander-polynomials-of-equivariant-slice-and-ribbon-knots-z8osmhdv91.pdf

Alexander polynomials of equivariant slice and ribbon knots in S^3

Murasugi discovered two criteria that must be satisfied by the Alexander polynomial of a periodic knot. We generalize these to the case of twisted Alexander polynomials. Examples demonstrate the application of these new criteria, including to knots with trivial Alexander polynomial, such as the two polynomial 1 knots with 11 crossings. 
Hartley found a restrictive condition satisfied by the Alexander polynomial of any freely periodic knot. We generalize this result to the twisted Alexander polynomial and illustrate the applicability of this extension in cases in which Hartley's criterion does not apply.

/pdf/twisted-alexander-polynomials-of-periodic-knots-2c4a5arkm9.pdf

We use the Heegaard-Floer homology correction terms defined by Ozsvath--Szabo to formulate a new obstruction for a knot to be of finite order in the smooth concordance group. This obstruction bears a formal resemblance to that of Casson and Gordon but is sensitive to the difference between the smooth versus topological category. As an application we obtain new lower bounds for the concordance order of small crossing knots.

Order in the concordance group and Heegaard Floer homology

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Ozsváth-Szabó and Rasmussen invariants of doubled knots

Twisted Alexander polynomials of periodic knots

Alexander polynomials of equivariant slice and ribbon knots in S^3

Twisted Alexander polynomials of periodic knots

Order in the concordance group and Heegaard Floer homology