Related papers: Contractibility of the Kakimizu complex and symmet…
Using a recent description of the geometric stability manifold, we show the geometric stability manifold associated to any smooth projective complex surface is contractible. We then use this result to demonstrate infinitely many new…
We show that every canonical Seifert surface is (up to isotopy) given by a knot diagram in which the (open) Seifert disks are pairwise disjoint.
A contractible simplicial complex is constructed that parametrizes different ways of representing a fixed one-dimensional homology class in a closed orientable surface by isotopy classes of systems of disjoint oriented simple closed curves.…
We introduce the notion of a contractible subshift. This is a strengthening of the notion of strong irreducibility, where we require that the gluings are given by a block map. We show that a subshift is a retract of a full shift if and only…
A virtual knot that has a homologically trivial representative $\mathscr{K}$ in a thickened surface $\Sigma \times [0,1]$ is said to be an almost classical (AC) knot. $\mathscr{K}$ then bounds a Seifert surface $F\subset \Sigma \times…
A marked strongly invertible knot is a triple $(K,h,\delta)$ of a knot $K$ in $S^3$, a strong inversion $h$ of $K$, and a subarc $\delta \subset \operatorname{Fix}(h)\cong S^1$ bounded by $\operatorname{Fix}(h)\cap K\cong S^0$. An invariant…
We consider homologically essential simple closed curves on Seifert surfaces of genus one knots in $S^3$, and in particular those that are unknotted or slice in $S^3$. We completely characterize all such curves for most twist knots: they…
We give a new proof, using comparatively simple techniques, of the Sullivan conjecture: the space of pointed maps from the classifying space of the cyclic group of order $p$ to any finite-dimensional CW complex $K$ is contractible.
Roberts proved that a family of alternating, arborescent, prime knots each have at least $2^{2n-1}$ distinct minimal genus Seifert surfaces, where $n$ is the genus of the knot in question. We give a subfamily of these knots that have…
This paper focuses on the Kakimizu complex of a hyperbolic knot $K$. We define a complex $IS_\ell(K)$ to study incompressible Seifert surfaces of genus at most $\ell$, and prove that it is connected and that its diameter admits a linear…
We show that any smooth one-dimensional link in the real projective three-plane is the fixed-point locus of a smooth symplectic surface in the complex projective three-plane which is invariant under complex conjugation. The degree of the…
Under a simple assumption on Seifert surfaces, we characterise knots whose stable topological 4-genus coincides with the genus.
The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of…
This paper proves that every oriented non-disk Seifert surface $F$ for a knot $K$ in $S^3$ is smoothly concordant to a Seifert surface $F^{\prime}$ for a hyperbolic knot $K^{\prime}$ of arbitrarily large volume. This gives a new and simpler…
We present a discrete Morse-theoretic method for proving that a regular CW complex is homeomorphic to a sphere. We use this method to define bisimplices, the cells of a class of regular CW complexes we call bisimplicial complexes. The…
Examples are given to show that some compact contractible 4-manifolds can be knotted in the 4-sphere. It is then proved that any finitely presented perfect group with a balanced presentation is a knot group for an embedding of some…
If $K$ is a fibered knot in a closed, oriented $3$--manifold $Y$ with fiber $F$, and $\widehat{HFK}(Y,K,[F], g(F)-1;\mathbb Z/2\mathbb Z)$ has rank $r$, then the monodromy of $K$ is freely isotopic to a diffeomorphism with at most $r-1$…
If a knot K has Seifert matrix V_K and has a prime power cyclic branched cover that is not a homology sphere, then there is an infinite family of non-concordant knots having Seifert matrix V_K.
For a given smooth $2$-knot in $S^4$, we relate the existence of a smooth Seifert hypersurface of a certain class to the existence of irreducible $ SU(2)$-representations of its knot group. For example, we see that any smooth $2$-knot…
In this paper we prove that if S is a smooth, irreducible, projective, rational, complex surface and D an effective, connected, reduced divisor on S, then the pair (S,D) is contractible if the log-Kodaira dimension of the pair is $-\infty$.…