Abstract. We investigate the question of the existence of a Lagrangian concordance between two Legendrian knots in R 3 . In particular, we give obstructions to a concordance from an arbitrary knot to the standard Legendrian unknot, in terms of normal rulings. We also place strong restrictions on knots that have concordances both to and from the unknot and construct an infinite family of knots with non-reversible concordances from the unknot. Finally, we use our obstructions to present a complete list of knots with up to 14 crossings that have Legendrian representatives that are Lagrangian slice.
We prove the existence of a polynomial invariant that satisfies the HOMFLY skein relation for links in a lens space. In the process we also develop a skein theory of toroidal grid diagrams in a lens space.(i) J p,q satisfies the skein relationAs usual, the links L + , L − , and L 0 differ only in a small neighborhood. The exact construction of these links in L(p, q) is made clear in the subsequent text and their construction is consistent with that in [13].Remark 1.2. In a large class of rational homology spheres, Kalfagianni has found a power series valued invariant of framed links that satisfies the Kauffman skein relation [12]. The ideas in the proof of Theorem 1.1 should also be capable of showing that this invariant provides a Kauffman polynomial for links in L(p, q).Remark 1.3. The polynomial given by Theorem 1.1 is used in [7] to prove an analogue of the Franks-Williams-Morton inequality in L(p, q) with a universally tight contact structure. This inequality exhibits a degree of the HOMFLY polynomial as an upper bound on the maximal self-linking number.We organize the paper as follows: In Section 2 we set notation, and will review constructions and results from [13] and [2,3]. In Section 3 we prove some homotopy results and develop the skein theory on grid diagrams for links in lens spaces. In Section 4 we prove our main results. We define a collection of trivial links and show that there is exactly one trivial link in each free homotopy class of links. Secondly, we show using a complexity function that if the power-series invariant is Laurent 3 polynomial valued on each trivial link, then the same is true for any link. In Section 5 we calculate the polynomial in some examples.
Abstract. We give criteria for an invariant of lens space links to bound the maximal self-linking number in certain tight contact lens spaces. Our result generalizes that given by Ng [18] for links in S 3 with the standard tight contact structure. As a corollary we extend the Franks-Williams-Morton inequality to the setting of lens spaces.
We study certain linear representations of the knot group that induce augmentations in knot contact homology. This perspective enhances our understanding of the relationship between the augmentation polynomial and the A-polynomial of a knot. For example, we show that for 2-bridge knots the polynomials agree and that this is never the case for (non-2-bridge) torus knots, nor for a family of 3-bridge pretzel knots. In addition, we obtain a lower bound on the meridional rank of the knot. As a consequence, our results give a new proof that torus knots and a family of pretzel knots have meridional rank equal to their bridge number.
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