We show if L is any link in S 3 whose Khovanov homology is isomorphic to the Khovanov homology of T (2, 6) then L is isotopic to T (2, 6). We show this for unreduced Khovanov homology with Z coefficients.
Regarding neighbor balance, we consider natural generalizations of $D$-complete Latin squares and Vatican squares from the finite to the infinite. We show that if $G$ is an infinite abelian group with $|G|$-many square elements, then it is possible to permute the rows and columns of the Cayley table to create an infinite Vatican square. We also construct a Vatican square of any given infinite order that is not obtainable by permuting the rows and columns of a Cayley table. Regarding orthogonality, we show that every infinite group $G$ has a set of $|G|$ mutually orthogonal orthomorphisms and hence there is a set of $|G|$ mutually orthogonal Latin squares based on $G$. We show that an infinite group $G$ with $|G|$-many square elements has a strong complete mapping; and, with some possible exceptions, infinite abelian groups have a strong complete mapping.
We show if L is any link in S 3 whose Khovanov homology is isomorphic to the Khovanov homology of T (2, 6) then L is isotopic to T (2, 6). We show this for unreduced Khovanov homology with Z coefficients.
We prove that for a fixed braid index there are only finitely many possible shapes of the annular Rasmussen d t invariant of braid closures. Applying the same perspective to the knot Floer invariant Υ K (t), we show that for a fixed concordance genus of K there are only finitely many possibilities for Υ K (t). Focusing on the case of 3-braids, we compute the Rasmussen s invariant and the annular Rasmussen d t invariant of all 3-braid closures. As a corollary, we show that the vanishing/non-vanishing of the ψ invariant is entirely determined by the s invariant and the self-linking number.
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