Khovanov Homology, Distributive Structure Homology and Applications to Knot Theory
Open AccessRack and quandle homology are (co)homology theories introduced by R. Fenn, C. Rourke and B. Sanderson and J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito, respectively for distributive structures diagrammatically from Reidemeister moves in classical knot theory. In 1992, R. Fenn, C. Rourke and B. Sanderson introduced the notion of a rack space which is the geometric realization of a rack chain complex, and it was modified by T. Nosaka in order to define homotopy invariants of classical knots and knotted surfaces.Knot theory is the study of embeddings of an $n$-dimensional sphere in an $(n+2)$-dimensional space. In 1925, E. Artin introduced the technique to build a knotted sphere in $4$-dimensional space using a classical knot in $3$-dimensional space. In 1965, E. C. Zeeman proved that the $\pm1$-twist spin of any $n$-dimensional knot $K \subset S^{n+2}$ is unknotted in $S^{n+3}.$ Y. Marumoto and Y. Nakanishi in 1991 provided an alternate proof of Zeeman's remarkable discovery by using the moving picture method. Meanwhile, periodicity is one of the most interesting and important concepts in mathematics pervading both science and nature. In classical knot theory, periodicity has been well studied. Higher-dimensional periodic knots are also introduced by R. N. Cruz in 1991.Rack homology can be generalized to homology theory for the set-theoretic Yang-Baxter equation which was introduced by J. S. Carter, M. Elhamdadi and M. Saito in 2004. On the other hand, Khovanov Homology is the homological invariant of classical knots which categorifies Jones polynomial, and the Jones polynomial can be described using a certain Yang-Baxter operator. Therefore, we anticipate there will be a relation between rack homology (or Yang-Baxter homology) and Khovanov Homology.In this dissertation, we first prove the conjecture of M. Niebrzydowski and J. H. Przytycki on annihilation of torsion in rack and quandle homology of a finite quasigroup quandle, and then we generalize the annihilation theorem to quandle extensions and non-connected quandles. Moreover, in analogy to the quandle homotopy invariant, we introduce the shadow homotopy invariant of a classical knot, and prove that for a finite connected quandle $X$ the shadow homotopy invariant is equal to the quandle homotopy invariant multiplied by $|X|.$Next, we show that a $\pm1$-twist spin of a knotted trivalent graph is not always unknotted in $\mathbb{R}^{4}$ and give a sufficient condition for the $\pm1$-twist spin of a given knotted trivalent graph to be unknotted. Furthermore, we explain how to build a periodic knotted torus in $\mathbb{R}^{4}$ using a given classical knot in $\mathbb{R}^{3},$ and discuss certain quandle extensions to color periodic knotted tori.Lastly, we focus on torsion in Khovanov Homology. Although presence of $\mathbb{Z}_{2}$-torsion in Khovanov homology is a very common phenomenon, torsion of orders other than two appears very seldom. We analyze Khovanov homology of twist deformations of torus links and show that counterexamples to the PS braid conjecture can be obtained in this procedure. Moreover, we provide some examples showing that the Khovanov homology of the flat $2$-cabling of a given knot may have interesting torsion subgroups such as $\mathbb{Z}_{2^{r}}.$This dissertation is comprised of the author's research papers written during his graduate study.
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