We compute the knot Floer homology of knots with at most 12 crossings, as well as the τ invariant for knots with at most 11 crossings, using the combinatorial approach described by Manolescu, Ozsvath and Sarkar. We review their construction, giving two examples that can be workout out by hand, and we explain some ideas we used to simplify the computation. We conclude with a discussion of knot Floer homology for small knots, and we formulate a conjecture about the behavior of knot Floer homology under mutation, paying especially close attention to the Kinoshita–Terasaka knot and its Conway mutant. Finally, we discuss a conjecture of Rasmussen on relationship between Khovanov homology and knot Floer homology, and observe that it is consistent with our calculations.

Computations of Heegaard-Floer knot homology

This is a survey article about knot Floer homology. We present three constructions of this invariant: the original one using holomorphic disks, a combinatorial description using grid dia- grams, and a combinatorial description in terms of the cube of resolutions. We discuss the geomet- ric information carried by knot Floer homology, and the connection to three- and four-dimensional topology via surgery formulas. We also describe some conjectural relations to Khovanov-Rozansky homology. Knot Floer homology is an invariant of knots and links in three-manifolds. It was introduced independently by Ozsvath-Szabo (OS04c) and Rasmussen (Ras03) around 2002. Since then it has grown into a large subject. Its importance lies in the fact that it contains information about several non-trivial geometric properties of the knot (genus, slice genus, fiberedness, effects of surgery, etc.) Furthermore, knot Floer homology is computable: There exist general algorithms that can calculate it for arbitrary knots. These algorithms tend to get slow as the complexity of the knot increases, but there are also different methods that can be applied to special classes of knots and give explicit

An introduction to knot floer homology

Knot theory is the embedding of one topological space into another topological space. Knot theory was first proposed as the flawed vortex model of atoms by Lord Kelvin, but was later proved to be incorrect. Some fundamental theorems of involved in knot theory include one involving Reidemeister moves (transforming knots into unknots) and knot invariants such as the Alexander Polynomial, Conway Polynomial and Jones Polynomial that originated in math and physics. It has applications in cell replication, where DNA must be unknotted and undergo a recombination process. Knots can also be used to improve encryption processes and increase security. Ellie O’Brien Knot Theory and its Applications April 26, 2016 2 / 4 Summary 1 Theorem: Two diagrams represent equivalent knots if and only if one can be transformed into the other by a finite sequence of Reidemeister moves. 2 The knot invariant polynomials are skein related which means they are identical at every point except at one crossing. Ellie O’Brien Knot Theory and its Applications April 26, 2016 3 / 4

/pdf/knot-theory-and-its-applications-3s7ym9m5t0.pdf

Knot Theory and Its Applications

We show that nontrivial classical pretzel knots L (p, q, r) are hyperbolic with eight exceptions which are torus knots. We find Conway polynomials of n-pretzel links using a new computation tree. As applications, we compute the genera of n-pretzel links using these polynomials and find the basket number of pretzel links by showing that the genus and the canonical genus of a pretzel link are the same.

/pdf/some-invariants-of-pretzel-links-2aeoi39x0e.pdf

Some invariants of Pretzel links

We discuss the relation between arc index, maximal Thurston--Bennequin number, and Khovanov homology for knots. As a consequence, we calculate the arc index and maximal Thurston--Bennequin number for all knots with at most 11 crossings. For some of these knots, the calculation requires a consideration of cables which also allows us to compute the maximal self-linking number for all knots with at most 11 crossings.

On arc index and maximal Thurston-Bennequin number

In 1996, Cromwell and Nutt [7] found an upper bound on the arc index which is related to the minimal crossing number and conjectured that the upper bound achieves the lowest possible index for alternating links. Conjecture. LetL be any prime link. Then(L)6 c(L)+2. Moreover this inequality is strict if and only if L is not alternating. In this paper, we dene a new diagram, called a wheel diagram, of a link and use

An upper bound of arc index of links

In [1], E. Appleboim introduced the notion of double dating linking class-P invariants of finite type for framed links with a fixed linking matrix P and showed that all Vassiliev link invariants are of finite type for any linking matrix and in [13], R. Trapp provided a necessary condition for a knot invariant to be a Vassiliev invariant by using twist sequences. In this paper we provide a necessary condition for a framed link invariant to be a DD-linking class-P invariant of finite type by considering sequence of links induced from a double dating tangle. As applications we give a generalization of R. Trapp's result to see whether a link invariant is a Vassiliev invariant or not and apply the criterion for all non-zero coefficients of the Jones, HOMFLY, Q-, and Alexander polynomial.

Vassiliev invariants and double dating tangles

Suppose that an n-dimensional closed real projective manifold M, n ≥ 2, develops into an affine space RPn − RPn − 1 for an (n − 1)-dimensional subspace RPn − 1 of the projective space RPn. Then either M is convex or affine or M admits a flat foliation with a transverse invariant Hilbert metric. Further, if the codimension of is n − 1, then M is convex. We prove this statement by a use of a variation of Carriere's discompacte, a measure of non-compactedness of an affine group acting on an affine space.

Real projective manifolds developing into an affine space

Vassiliev invariants and knot polynomials

We calculate the Zulli's matrix of a periodic knot and give some necessary conditions for the Jones polynomial of a periodic knot, which are slightly different from Yokota's result.

/pdf/jones-polynomials-of-periodic-knots-1y3msn45cx.pdf

Chan-Young Park

Papers

An upper bound of arc index of links

Vassiliev invariants and double dating tangles

Real projective manifolds developing into an affine space

Vassiliev invariants and knot polynomials

Jones polynomials of periodic knots