arXiv subjects

Tamás Kálmán

Publications and source records attributed to Tamás Kálmán.

27 records · Page 2Linked to original sources

arXiv2012

Legendrian knots and exact Lagrangian cobordisms

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 $L\subset X$ which agrees with cylinders over Legendrian links $Λ_+$ and $Λ_-$ at the positive and negative ends induces a differential graded algebra (DGA) map from the Legendrian contact homology DGA of $Λ_+$ to that of $Λ_-$. 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.

math.SG↗

arXiv2011

Sutured Floer homology and hypergraphs

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 support of the sutured Floer homology of the sutured manifold complementary to F is affine isomorphic to the set of lattice points given as hypertrees in a certain hypergraph that is naturally associated to the diagram. This implies that the Floer groups in question are supported in a set of Spin^c structures that are the 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, together with work in progress of the second author and others, our correspondence suggests 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.

math.GT↗

arXiv2011

A version of Tutte's polynomial for hypergraphs

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. In the definition, we associate activities to hypertrees, which are generalizations of the indicator function of the edge set of a spanning tree. We prove that hypertrees form a lattice polytope which is the set of bases in a polymatroid. In fact, we extend our invariants to integer polymatroids as well. 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.

math.CO↗

arXiv2009

Inner products on the Hecke algebra of the braid group

We point out that the Homfly 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. In these structures, the sets of positive, respectively negative permutation braids become orthonormal bases. In the second case, many inner products can be geometrically interpreted through Legendrian fronts and rulings.

math.GT↗

arXiv2008

Meridian twisting of closed braids and the Homfly polynomial

Let $β$ be a braid on $n$ strands, with exponent sum $w$. Let $Δ$ be the Garside half-twist braid. We prove that the coefficient of $v^{w-n+1}$ in the Homfly polynomial of the closure of $β$ agrees with $(-1)^{n-1}$ times the coefficient of $v^{w+n^2-1}$ in the Homfly polynomial of the closure of $βΔ^2$. This coincidence implies that the lower Morton--Franks-Williams estimate for the $v$--degree of the Homfly polynomial of $\hatβ$ is sharp if and only if the upper MFW estimate is sharp for the $v$--degree of the Homfly polynomial of $\hat{βΔ^2}$.

math.GT↗

arXiv2007

Rulings of Legendrian knots as spanning surfaces

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) knots, we prove 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 (a.k.a. diagram genus) of any knot is an upper bound for the genera of its 2-graded rulings.

math.GT↗

arXiv2006

Maximal Thurston-Bennequin number of +adequate links

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.

math.GT↗

arXiv2006

Braid-positive Legendrian links

Any link that is the closure of a positive braid has a natural Legendrian representative. These were introduced in an earlier paper, where their Chekanov--Eliashberg contact homology was also evaluated. In this paper we re-phrase 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.

math.SG↗

arXiv2004

One parameter families of Legendrian torus knots

We compute the Chekanov-Eliashberg contact homology of what we call the Legendrian closure of a positive braid. We also construct an augmentation for each such link diagram. Then we apply the monodromy techniques established in an earlier paper to a certain natural loop in the space L' of positive Legendrian (p,q) torus knots to show that L' has a nontrivial fundamental group, which is mapped non-injectively into the fundamental group of K by the map induced by inclusion (here, K is the space of all smooth positive (p,q) torus knots). This is the first example that Legendrian and classical knots behave differently at the level of one parameter families.

math.GT↗