Language：English   Publishing type：Research paper (scientific journal)   Publisher：EUROPEAN MATHEMATICAL SOC  

We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair (X, L) consisting of an exact symplectic manifold X and an exact Lagrangian cobordism L subset of X which agrees with cylinders over Legendrian links Lambda(+) and Lambda (-) at the positive and negative ends induces a differential graded algebra (DGA) map from the Legendrian contact homology DGA of Lambda(+) to that of Lambda (-) .We give a gradient flow tree description of the DGA maps for certain pairs (X, L), which in turn yields a purely combinatorial description of the cobordism map for elementary cobordisms, i.e., cobordisms that correspond to certain local modifications of Legendrian knots. As an application, we find exact Lagrangian surfaces that fill a fixed Legendrian link and are not isotopic through exact Lagrangian surfaces.

DOI： 10.4171/JEMS/650

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

Tutte's dichromate T(x, y) is a well known graph invariant. Using the original definition in terms of internal and external activities as our point of departure, we generalize the valuations T(x, 1) and T(1, y) to hypergraphs. Our generating functions are sums over hypertrees, i.e., instances of a certain generalization of the indicator function of the edge set of a spanning tree. We prove that hypertrees are the lattice points in a polytope which in turn is the set of bases in a polymatroid. In fact, we extend our invariants to integer polymatroids as well. Several properties are established, including a generalization of the deletion contraction formulas. We also examine hypergraphs that can be represented by planar bipartite graphs, write their hypertree polytopes in the form of a determinant, and prove a duality property that leads to an extension of Tutte's Tree Trinity Theorem. (C) 2013 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.aim.2013.06.001

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：INT PRESS BOSTON, INC  

By applying Seifert's algorithm to a special alternating diagram of a link L, one obtains a Seifert surface F of L. We show that the set of Spin(c) structures that support the sutured Floer homology of the sutured manifold complementary to F is affine isomorphic to the set of hypertrees in a certain hypergraph that is naturally associated to the diagram. This implies that the support in question is the set of integer lattice points of a convex polytope. This property has an immediate extension to Seifert surfaces arising from homogeneous link diagrams (including all alternating and positive diagrams).In another direction, our results and work in progress of the second author with Murakami and Postnikov suggest a method for computing the "top" coefficients of the HOMFLY polynomial of a special alternating link from the sutured Floer homology of a Seifert surface complement for a certain dual link.

DOI： 10.4310/MRL.2012.v19.n6.a11

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER  

We claim that the Homily polynomial (that is to say, Ocneanu's trace functional) contains two polynomial-valued inner products on the Hecke algebra representation of Artin's braid group. These bear a close connection to the Morton-Franks-Williams inequality. With respect to these structures, the set of positive, respectively negative permutation braids becomes an orthonormal basis. In the second case, many inner products can be geometrically interpreted through Legendrian fronts and rulings. (C) 2010 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.topol.2010.12.014

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：CAMBRIDGE UNIV PRESS  

Let beta be a braid on n strands, with exponent sum w. Let Delta be the Garside half-twist braid. We prove that the coefficient of v(w-n+1) in the Homfly polynomial of the closure of beta agrees with (-1)(n-1) times the coefficient of v(w+n2-1) in the Homfly polynomial of the closure of beta Delta(2). This coincidence implies that the lower Morton-Franks-Williams estimate for the v-degree of the Homfly polynomial of (beta) over cap is sharp if and only if the upper MFW estimate is sharp for the v-degree of the Homfly polynomial of (beta Delta(2)) over cap.

DOI： 10.1017/S0305004108002016

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：INT PRESS BOSTON, INC  

We construct a Legendrian 2-torus in the 1-jet space of S-1 x R ( or of R-2) from a loop of Legendrian knots in the 1-jet space of R. The differential graded algebra (DGA) for the Legendrian contact homology of the torus is explicitly computed in terms of the DGA of the knot and the monodromy operator of the loop. The contact homology of the torus is shown to depend only on the chain homotopy type of the monodromy operator. The construction leads to many new examples of Legendrian knotted tori. In particular, it allows us to construct a Legendrian torus with DGA which does not admit any augmentation (linearization) but which still has non-trivial homology, as well as two Legendrian tori with isomorphic linearized contact homologies but with distinct contact homologies.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PACIFIC JOURNAL MATHEMATICS  

Each ruling of a Legendrian link can be naturally treated as a surface. For knots, the ruling is 2-graded if and only if the surface is orientable. For 2-graded rulings of homogeneous (in particular, alternating and positive) knots, we show that the genus of this surface is at most the genus of the knot. While this is not true in general, we do prove that the canonical genus of any knot is an upper bound for the genera of its 2-graded rulings.

DOI： 10.2140/pjm.2008.237.287

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

The class of +adequate links contains both alternating and positive links. Generalizing results of Tanaka (for the positive case) and Ng (for the alternating case), we construct fronts of an arbitrary +adequate link A so that the diagram has a ruling; therefore its Thurston-Bennequin number is maximal among Legendrian representatives of A. We derive consequences for the Kauffman polynomial and Khovanov homology of +adequate links.

DOI： 10.1090/S0002-9939-08-09478-1

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Any link that is the closure of a positive braid has a natural Legendrian representative. These representatives were introduced in an earlier paper, where their Chekanov-Eliashberg contact homology was also evaluated. In this paper, we rephrase and improve that computation using a matrix representation. In particular, we present a way of finding all augmentations of such Legendrians, construct an augmentation which is also a ruling, and find surprising links to LU -decompositions and Gröbner bases.

DOI： 10.1155/IMRN/2006/14874

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We consider S1 -families of Legendrian knots in the standard contact R3. We define the monodromy of such a loop, which is an automorphism of the Chekanov-Eliashberg contact homology of the starting (and ending) point. We prove this monodromy is a homotopy inhvariant of the loop (Theorem 1.1). We also establish techniques to address the issue of Reidemeister moves of Lagrangian projections of Legendrian links. As an application, we exhibit a loop of right-handed Legendrian torus knots which is non-contractible in the space Leg(S1,R3) of Legendrian knots, although it is contractible in the space Emb(S1,R3) of smooth knots. For this result, we also compute the contact homology of what we call the Legendrian closure of a positive braid (Definition 6.1) and construct an augmentation for each such link diagram. © Geometry & Topology Publications.

DOI： 10.2140/gt.2005.9.2013

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DOI： 10.1016/S0166-8641(01)00176-6

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In this paper we investigate Σ1,0-maps of closed surfaces into the plane, specifically, the singular sets of such maps. This set is the disjoint union of finitely many embedded circles in the surface; we will determine all possible numbers of components for each surface. During this survey we will construct singular maps of all closed surfaces into the plane which are simplest in the sense that they have the least possible number of cusps (0 or 1) and under this condition their singular sets have the least possible number of components (1 or 2). Additionally, we will provide a simplified and shortened proof of the dimension 2 case of the theorem concerning the elimination of cusps (due to Millett, and Levine for the higher-dimensional cases). © 2000 Elsevier Science B.V. All rights reserved.

DOI： 10.1016/s0166-8641(99)00105-4

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