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Torsion in Khovanov homology of semi-adequate links

100%
Przytycki J., Sazdanović R.
Fundamenta Mathematicae
|
2014
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tom 225
|
nr 1
277-303
EN
The goal of this paper is to address A. Shumakovitch's conjecture about the existence of ℤ₂-torsion in Khovanov link homology. We analyze torsion in Khovanov homology of semi-adequate links via chromatic cohomology for graphs, which provides a link between link homology and the well-developed theory of Hochschild homology. In particular, we obtain explicit formulae for torsion and prove that Khovanov homology of semi-adequate links contains ℤ₂-torsion if the corresponding Tait-type graph has a cycle of length at least 3. Computations show that torsion of odd order exists but there is no general theory to support these observations. We conjecture that the existence of torsion is related to the braid index.
2
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Burnside kei

100%
Niebrzydowski M., Przytycki J.
Fundamenta Mathematicae
|
2006
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tom 190
|
nr 1
211-229
EN
This paper is motivated by a general question: for which values of k and n is the universal Burnside kei Q̅(k,n) finite? It is known (starting from the work of M. Takasaki (1942)) that Q̅(2,n) is isomorphic to the dihedral quandle Zₙ and Q̅(3,3) is isomorphic to Z₃ ⊕ Z₃. In this paper, we give a description of the algebraic structure for Burnside keis Q̅(4,3) and Q̅(3,4). We also investigate some properties of arbitrary quandles satisfying the universal Burnside relation a = ⋯ a∗b∗ ⋯ ∗a∗b. Invariants of links related to the Burnside kei Q̅(k,n) are invariant under n-moves.
3
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Cocycle invariants of codimension 2 embeddings of manifolds

100%
Przytycki J., Rosicki W.
Banach Center Publications
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2014
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tom 103
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nr 1
251-289
EN
We consider the classical problem of a position of n-dimensional manifold Mⁿ in $ℝ^{n+2}$. We show that we can define the fundamental (n+1)-cycle and the shadow fundamental (n+2)-cycle for a fundamental quandle of a knotting $Mⁿ → ℝ^{n+2}$. In particular, we show that for any fixed quandle, quandle coloring, and shadow quandle coloring, of a diagram of Mⁿ embedded in $ℝ^{n+2}$ we have (n+1)- and (n+2)-(co)cycle invariants (i.e. invariant under Roseman moves).
4
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Torsion in graph homology

81%
Helme-Guizon L., Przytycki J., Rong Y.
Fundamenta Mathematicae
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2006
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tom 190
|
nr 1
139-177
EN
Khovanov homology for knots has generated a flurry of activity in the topology community. This paper studies the Khovanov type cohomology for graphs with a special attention to torsion. When the underlying algebra is ℤ[x]/(x²), we determine precisely those graphs whose cohomology contains torsion. For a large class of algebras, we show that torsion often occurs. Our investigation of torsion led to other related general results. Our computation could potentially be used to predict the Khovanov-Rozansky sl(m) homology of knots (in particular (2,n) torus knot). We also predict that our work is connected with Hochschild and Connes cyclic homology of algebras.
5
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Signature of rotors

81%
Dąbkowski M., Ishiwata M., Przytycki J., Yasuhara A.
Fundamenta Mathematicae
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2004
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tom 184
|
nr 1
79-97
EN
Rotors were introduced as a generalization of mutation by Anstee, Przytycki and Rolfsen in 1987. In this paper we show that the Tristram-Levine signature is preserved by orientation-preserving rotations. Moreover, we show that any link invariant obtained from the characteristic polynomial of the Goeritz matrix, including the Murasugi-Trotter signature, is not changed by rotations. In 2001, P. Traczyk showed that the Conway polynomials of any pair of orientation-preserving rotants coincide. We show that there is a pair of orientation-reversing rotants with different Conway polynomials.
6
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Torsion in one-term distributive homology

81%
Crans A., Przytycki J., Putyra K.
Fundamenta Mathematicae
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2014
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tom 225
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nr 1
75-94
EN
The one-term distributive homology was introduced in [Prz] as an atomic replacement of rack and quandle homology, which was first introduced and developed by Fenn-Rourke-Sanderson [FRS] and Carter-Kamada-Saito [CKS]. This homology was initially suspected to be torsion-free [Prz], but we show in this paper that the one-term homology of a finite spindle may have torsion. We carefully analyze spindles of block decomposition of type (n,1) and introduce various techniques to compute their homology precisely. In addition, we show that any finite group can appear as the torsion subgroup of the first homology of some finite spindle. Finally, we show that if a shelf satisfies a certain, rather general, condition then the one-term homology is trivial-this answers a conjecture from [Prz] affirmatively.
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