arXiv subjects
Publications and source records attributed to David Popović.
We introduce a variant of reduced Khovanov-Rozansky homology defined in terms of the variables alpha_1, ..., alpha_{n-1} that correspond to the regions between the strands. We rigorously reestablish several folklore results in terms of our framework, including the equivalence between Khovanov's and Rasmussen's approaches. Furthermore, we show that Rasmussen's chain complex is a free resolution of the Rouquier chain complex.
We define a 'full' version of Khovanov-Rozansky homology: a chain complex CH(K) over F[U,V] whose chain homotopy type is a knot invariant and which has the property that setting U = V = 0 recovers the reduced Khovanov-Rozansky homology. We explore the algebraic structure of CH(K) and show that a variation of the structure theorem for knot Floer homology applies. This allows us to define counterparts to knot Floer concordance invariants $τ$, $ε$ and $ϕ_j$ in the Khovanov-Rozansky setting.
This paper classifies the chain homotopy equivalence types of knot Floer complexes $CFK_{\mathbb{F}[U,V]}(K)$ of knot Floer width 2. They have no nontrivial local systems. As an application, this shows that all Montesinos knots admit a basis that can be simultaneously horizontally and vertically simplified.
We classify isomorphism and chain homotopy equivalence classes of finitely generated $\mathbb{Z} \oplus \mathbb{Z}$ graded free chain complexes over $\frac{\mathbb{F}[U,V]}{(UV)}$. As a corollary, we establish that every link Floer complex $CFL(Y, L)$ over the ring $\frac{\mathbb{F}[U,V]}{(UV)}$ splits uniquely as a direct sum of snake complexes and local systems. This generalizes and extends the results of Petkova and Dai, Hom, Stoffregen, and Truong. We give the first example of an essentially infinite knot Floer-like complex, i.e., a complex satisfying all formal properties of link Floer complexes of knots in $S^3$ and whose chain homotopy equivalence class does not admit a representative of the form $C \otimes_{\mathbb{F}} \mathbb{F}[U,V]$. Finally, we also describe the first example of a knot Floer-like complex that does not admit a simultaneously vertically and horizontally simplified basis.
We study algebraic obstructions to realizability of local equivalence classes of knot-like complexes. We classify local equivalence classes of knot-like complexes over $\mathbb{F}[U,V]$, answering a question of Dai, Hom, Stoffregen and Truong.
This expository article gives a thorough and well-motivated account of the proof of the nilpotence theorem by Devinatz-Hopkins-Smith.
For each element $u$ in a finite group $G$ define a map $θ_u\colon G\to G$ by $θ_u(g)=[g^{-u},g]$ and set $Θ_G(u)=\{g\in G\mid θ_u^n(g)=g \hbox{ for some } n>0\}$. Then $θ_u$ induces a permutation of $Θ_G(u)$; let $β_G(u)$ be the number of orbits apart from $\{1\}$. Building on work of J.N. Bray, R.A. Wilson and the second author, we show that the index of the soluble radical of a finite group $G$ is bounded in terms of the values of $β_G(u)$ for $2$-elements $u$.
Using the computer algebra program GAP, we show that all crystallographic groups in dimensions at most 4 are distinguished from each other by their sets of finite quotients.