We give a new combinatorial construction of the hat version of the link Floer homology over Z/2Z and verify that in many examples, our complex is smaller than Manolescu–Ozsváth–Sarkar one. Introduction Knot Floer homology is a powerful knot invariant constructed by Ozsváth–Szabo [11] and Rasmussen [14]. In its basic form, the knot Floer homology ĤFK(K) of a knot K ∈ S is a finite–dimensional bi...

Twist knots form a family of special two–bridge knots which include the trefoil knot and the figure eight knot. The knot group of a two–bridge knot has a particularly nice presentation with only two generators and a single relation. One could find our interest in this family of knots in the following facts: first, twist knots except the trefoil knot are hyperbolic; and second, twist knots are n...

We shall explain how knot, link and tangle enumeration problems can be expressed as matrix integrals which will allow us to use quantum field-theoretic methods. We shall discuss the asymptotic behaviors for a great number of intersections. We shall detail algorithms used to test our conjectures. 1. Classification and Enumeration of Knots, Links, Tangles A knot is defined as a closed, non-self-i...

Given a knot K in the 3-sphere, let QK be its fundamental quandle as introduced by D. Joyce. Its first homology group is easily seen to be H1(QK) ∼= Z. We prove that H2(QK) = 0 if and only if K is trivial, and H2(QK) ∼= Z whenever K is non-trivial. An analogous result holds for links, thus characterizing trivial components. More detailed information can be derived from the conjugation quandle: ...

We derive, from the A-polynomial of a knot, a single variable polynomial for the knot, called C-polynomial, and explore topological and geometrical information about the knot encoded in the C-polynomial.  2003 Elsevier B.V. All rights reserved. MSC: 57N10; 57M25; 57M27; 57M40

This is a survey talk on one of the best known quantum knot invariants, the colored Jones polynomial of a knot, and its relation to the algebraic/geometric topology and hyperbolic geometry of the knot complement. We review several aspects of the colored Jones polynomial, emphasizing modularity, stability and effective computations. The talk was given in the Mathematische Arbeitstagung June 24-J...

Two fundamental theorems of classical knot theory, by J. Alexander, are that every knot is a closed braid, and second, that a certain procedure (see [1,2] for both results) assigns to each knot K a polynomial A(K) in T, which only depends on the topological type of the knot once normalized by a power of T to take a positive value at T = 0. With this normalization, take T to be a transcendental ...

The presence of a fractional-exponent growth law relating knot energy and knot topology is known to be an essential characteristic for the existence of ‘ideal’ knots. In this paper, we show that the energy infimum EN stratified at the Hopf charge N of the knot energy of the Faddeev type induced from the Hopf fibration S/S (nR1) in general dimensions obeys the sharp fractional-exponent growth la...

We suggest a method to detect that two periodic knots are not equivariantly concordant, using surgery on factor links. We construct examples which satisfy all known necessary conditions for equivariant slice knots — Naik’s and Choi-Ko-Song’s improvements of classical results on Seifert forms and Casson-Gordon invariants of slice knots — but are not equivariantly slice. Introduction An oriented ...

We resolve a question of Fuchs and Tabachnikov by showing that there is a Legendrian knot in standard contact R with zero Maslov number which is not Legendrian isotopic to its mirror. The proof uses the differential graded algebras of Chekanov. A Legendrian knot in standard contact R is a knot which is everywhere tangent to the two-plane distribution induced by the contact one-form dz− y dx. Tw...