In the present paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to the computation of the splitting number. Finally, we use the slice-torus link invariants, and the Whitehead doubling to define new strong concordance invariants for links, which are proven to be independent from the corresponding slice-torus link invariant.2010 Mathematics Subject Classification. 57M27. 1 These function were originally defined only for integer-valued slice-torus invariants. Of course, the same definition works for all slice-torus invariants, and most of the properties proved in [23] still hold. 1 2 ALBERTO CAVALLO AND CARLO COLLARIwhere ν is a slice-torus invariant and W ± (K, t) denotes the positive (resp. negative) ttwisted Whitehead double of K. These functions are non-increasing, non-constant, take values respectively in [0, 1] and [−1, 0], and assume both the maximal and the minimal possible values. In particular, if the slice-torus invariant is integer-valued all the information contained in each function can be condensed into a single integer. These integers, denoted by t ν and t ν , are defined as the maximal value of t such that F ν (K, t) and F ν (K, t), respectively, assume their maximum. It is not difficult to see that t ν (K) = −t ν (−K * ) − 1, where −K * is the mirror image of K with the orientation reversed, so these invariants contain the same amount of information. At the time of writing it is still unknown whether the invariant t ν can provide new information with respect to ν. In fact there are some hints in the opposite direction; for instance, it is known that t τ = 2τ − 1 ([15, Theorem 1.5]) and it has been conjectured that t s/2 = 3s/2 − 1 ([29]).The aim of the present paper is to extend these definitions and constructions to the case of links, and to describe some applications and examples. Before stating the main results of this paper let us recall a few basic facts about link concordance. The first thing one should point out is that the definition of concordance is no longer unique. Two oriented links in S 3 are said to be ⊲ weakly concordant if there exists a genus 0 connected, compact, oriented surface, properly embedded in S 3 × [0, 1], bounding the two links; ⊲ strongly concordant if there exists a disjoint union of annuli, properly embedded in S 3 × [0, 1], such that each of them bounds a component of each link. In particular, strongly concordant links should have the same number of components. A link is said to be weakly (resp. strongly) slice if is weakly (resp. strongly) concordant to an unlink. Similarly, one can define a (weak ) slice genus and a strong slice genus. The former is just the minimal genus of any connected, compact, oriented surface properly embedded in S 3 × [0, 1] bounding the link. The latter has a similar definition but one has t...
In this article we introduce a family of transverse invariants arising from the deformations of Khovanov homology. This family includes the invariants introduced by Plamenevskaya and by Lipshitz, Ng, and Sarkar. Then, we investigate the invariants arising from Bar-Natan's deformation. These invariants, called β-invariants, are essentially equivalent to Lipshitz, Ng, and Sarkar's invariants ψ ± . From the β-invariants we extract two non-negative integers which are transverse invariants (the c-invariants). Finally, we give several conditions which imply the non-effectiveness of the c-invariants, and use them to prove several vanishing criteria for the Plamenevskaya invariant [ψ], and the non-effectiveness of the vanishing of [ψ], for all prime knots with less than 12 crossings. Theorem 1.1 (Bennequin [3], Orevkov and Shevchishin [19], Wrinkle [27]). Any transverse link is transversely isotopic to the closure of a braid (with axis the z-axis). Moreover, two braids represent the same transverse link if and only if they are related by a finite sequence of braid relations, conjugations, positive stabilizations, and positive destabilizations 3 . These moves are called transverse Markov moves. Remark 1.3. Braids are naturally oriented, and their orientation coincide with the orientation of the corresponding transverse link. Remark 1.4. By adding the negative stabilization to the set of transverse Markov moves one recovers the full set of Markov moves.
Multipath cohomology is a cohomology theory for directed graphs, which is defined using the path poset. The aim of this paper is to investigate combinatorial properties of path posets and to provide computational tools for multipath cohomology. In particular, we develop acyclicity criteria and provide computations of multipath cohomology groups of oriented linear graphs. We further interpret the path poset as the face poset of a simplicial complex, and we investigate realisability problems.
In this paper we introduce a new cohomology theory for (directed) graphs, which we call multipath cohomology. Our construction interpolates between the chromatic homology, introduced by L. Helme-Guizon and Y. Rong, and the homology for directed graphs introduced by P. Turner and E. Wagner. We prove that the multipath cohomology satisfies some functorial properties and we describe its connection with chromatic homology. We provide a number of sample computations showing that: the multipath cohomology is different from Turner-Wagner's and chromatic homologies, it does not vanish on trees, and, when evaluated at the coherent polygon, it recovers the Hochschild homology. We conclude by presenting some open questions and related problems.
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