arXiv subjects
Publications and source records attributed to Paula Truöl.
For every $n \geq 4$, we demonstrate the existence of non-isotopic, smooth $(n-2)$-knots in $S^n$ with diffeomorphic traces. We give two proofs: the first by generalising the RBG link construction to all dimensions, and the second as an application of work of Plotnick. Conversely, we prove that for every $n \geq 4$, the unknot in $S^n$ is detected by the diffeomorphism type of its surgery and hence of its trace.
We examine how the Rasmussen invariant, satellite operations, and null-homologous twists can be used to establish infinite order of knots in the smooth concordance group. As an application, we show that the Conway knot has infinite concordance order.
T-positive links form a subset of strongly quasipositive links that strictly contains the set of all non-split braid positive links. Analogous to Baader's characterisation of positive links as precisely the strongly quasipositive and homogeneous links, we show that T-positive links are precisely the strongly quasipositive links that are the closures of T-homogeneous braids. This complements previous characterizations of T-positive links by Rudolph and Banfield as links arising as boundaries of positive Hopf-plumbed baskets, or closures of staircase braids. We examine the behavior of T-positive links under cabling operations and connected sums, and demonstrate that all strongly quasipositive, fibered knots with at most 12 crossings are T-positive. Additionally, we compare T-positivity with other positivity notions for links and compile open questions.
We show that every non-trivial strongly quasipositive link is smoothly concordant to infinitely many pairwise non-isotopic strongly quasipositive links. In contrast to our result, Baker conjectured that smoothly concordant strongly quasipositive fibered knots are isotopic. Our construction uses a satellite operation whose companion is a slice knot with maximal Thurston-Bennequin number -1.
In this short note, we prove that every closed, oriented, connected 3-manifold arises as Dehn surgery along a braid positive link.
We show that for every genus $g \geq 0$, there exist quasipositive knots $K_0^g$ and $K_1^g$ such that there is a cobordism of genus $g=|g_4(K_1^g)-g_4(K_0^g)|$ between $K_0^g$ and $K_1^g$, but there is no ribbon cobordism of genus $g$ in either direction and thus no complex cobordism between these two knots. This gives a negative answer to a question posed by Feller in 2016.
We show that there exists an algorithm that takes as input two closed, simply connected, topological 4-manifolds and decides whether or not these 4-manifolds are homeomorphic. In particular, we explain in detail how closed, simply connected, topological 4-manifolds can be naturally represented by a Kirby diagram consisting only of 2-handles. This representation is used as input for our algorithm. Along the way, we develop an algorithm to compute the Kirby-Siebenmann invariant of a closed, simply connected, topological 4-manifold from any of its Kirby diagrams and describe an algorithm that decides whether or not two intersection forms are isometric. In a slightly different direction, we discuss the decidability of the stable classification of smooth manifolds with more general fundamental groups. Here we show that there exists an algorithm that takes as input two closed, oriented, smooth 4-manifolds with fundamental groups isomorphic to a finite group with cyclic Sylow 2-subgroup, an infinite cyclic group, or a group of geometric dimension at most 3 (in the latter case we additionally assume that the universal covers of both 4-manifolds are not spin), and decides whether or not these two 4-manifolds are orientation-preserving stably diffeomorphic.
We determine the locally flat cobordism distance between torus knots with small and large braid index, up to high precision. Here small means 2, 3, 4, or 6. As an application, we derive a surprising fact about torus knots that appear as cross-sections of almost minimal cobordisms between two-stranded torus knots and the trivial knot.
The nonorientable four-ball genus of a knot $K$ in $S^3$ is the minimal first Betti number of nonorientable surfaces in $B^4$ bounded by $K$. By amalgamating ideas from involutive knot Floer homology and unoriented knot Floer homology, we give a new lower bound on the smooth nonorientable four-ball genus $γ_4$ of any knot. This bound is sharp for several families of torus knots, including $T_{4n,(2n\pm 1)^2}$ for even $n\ge 2$, a family Longo showed were counterexamples to Batson's conjecture. We also prove that, whenever $p$ is an even positive integer and $\frac{p}{2}$ is not a perfect square, the torus knot $T_{p,q}$ does not bound a locally flat Möbius band for almost all integers $q$ relatively prime to $p$.
We classify 3-braid knots whose topological 4-genus coincides with their Seifert genus, using McCoy's twisting method and the Xu normal form. In addition, we give upper bounds for the topological 4-genus of positive and strongly quasipositive 3-braid knots.
We provide explicit formulas for the integer-valued smooth concordance invariant $\upsilon(K) = Υ_K(1)$ for every 3-braid knot $K$. We determine this invariant, which was defined by Ozsváth, Stipsicz and Szabó, by constructing cobordisms between 3-braid knots and (connected sums of) torus knots. As an application, we show that for positive 3-braid knots $K$ several alternating distances all equal the sum $g(K) + \upsilon(K)$, where $g(K)$ denotes the 3-genus of $K$. In particular, we compute the alternation number, the dealternating number and the Turaev genus for all positive 3-braid knots. We also provide upper and lower bounds on the alternation number and dealternating number of every 3-braid knot which differ by 1.