arXiv subjects
Publications and source records attributed to Sebastian Baader.
We derive an almost sharp lower bound on the Casson invariant of positive braid knots. As a consequence, we show that the $\Delta$-unknotting number is quasi-additive on the set of positive and negative braids on a fixed number of strands.
In autonomous racing, especially in competitions such as Formula Student Driverless, precise planning of the target velocity of a race car is crucial for competitive lap times and stable driving behavior. Especially at high speeds, Velocity Planning (VP) is a significant challenge as it has to be performed in real time, taking into account track layouts, environmental influences, mechanical tolerances, and the resulting control inaccuracies. In this paper, we present a novel approach to VP that dynamically adapts to such changing conditions. Instead of estimating the physical Tire-Road Friction Coefficient (TRFC), a continuous scaling factor is inferred indirectly from vehicle stability. This factor not only reflects the effective tire-road interaction but also captures effects of control inaccuracies. From this, we generate a continuous friction map, which serves as a robust, adaptive basis for computing the optimal target speed, accounting for both vehicle and environmental limits. Our proposed approach was evaluated on a real Formula Student race car, showing a lap time improvement of 35 % over ten laps and an average increase of 8 % compared to a non-adaptive approach.
Autonomous race cars, such as in Formula Student Driverless, operate close to their physical handling limits. The resulting highly nonlinear vehicle behavior increases the path tracking complexity, especially on narrow tracks. Model Predictive Control (MPC) is commonly used to address this issue, a method whose performance is closely tied to the accuracy of the underlying prediction model. This paper presents a novel, real-time capable prediction model for autonomous race cars that adjusts to changing conditions by combining information from past runs and the current driving situation. Our model is divided into three consecutive submodels: a nominal Kinematic Bicycle Model, an offline Bayesian Linear Regression (BLR) model, and an online Sparse Gaussian Process Regression (SGPR) model. The proposed approach enables efficient integration of all available data without significantly increasing computational cost, ensuring high prediction accuracy and a quantitative uncertainty assessment right from the start of the run. Compared to existing approaches, an improvement in prediction accuracy of up to 57% was achieved. Further, we successfully demonstrated the practical applicability of the model within an MPC-based path tracking controller on a real Formula Student race car.
We determine the crossing number of polynomial size curve systems on standard surfaces, in terms of the genus, up to high precision.
A system of simple closed curves on a surface of genus $g$ is said to be sparse if their average pairwise intersection number does not exceed one. We show that the maximal size of a sparse curve systems grows roughly like a function of type $c^{\sqrt{g}}$, with $c$ between $2$ and $81938$.
We determine the maximal number of systoles among all spheres with $n$ punctures endowed with a complete Riemannian metric of finite area.
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.
We construct efficient topological cobordisms between torus links and large connected sums of trefoil knots. As an application, we show that the signature invariant $\sigma_\omega$ at $\omega=\zeta_6$ takes essentially minimal values on torus links among all concordance homomorphisms with the same normalisation on the trefoil knot.
We show that the family of systoles of hyperbolic surfaces associated with congruence lattices in $\mathrm{SL}_2(\mathbb{Z})$ have asymptotically minimal crossing number.
We determine the cobordism distance between all positive 3-braid links and connected sums of trefoil knots, up to a constant error. This may be seen as a reinterpretation of Coxeter's finiteness result on the braid group on three strands modulo third powers of the generators. We go on to explore the limits of Coxeter's result, and find the cobordism distance between three strand torus links and almost all connected sums of two strand torus links, up to a constant error.
We solve an asymptotic variant of a smooth version of the $A_n$-realization problem for plane curves. As an application, we determine the cobordism distance between torus links of type $T(d,d)$ and $T(2,N)$ up to an error of at most $3d$. We also discuss the limits of knot theoretic approaches aimed at solving the $A_n$-realization problem.
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 prove that fibred knots cannot be untied with $\bar{t}_{2k}$-moves, for all $k \geq 2$. More generally, we give an upper bound on the number of two strand twist operations that allow to untie a knot with non-trivial HOMFLY polynomial, in terms of the minimal crossing number, and the braid index. As a by-product, we prove that the braid index of a two-bridge knot cannot be lowered by applying $t_{2k}$-moves, for all but finitely many $k \in \mathbb{N}$.
For all natural numbers $N$ and prime numbers $p$, we find a knot $K$ whose skein polynomial $P_K(a,z)$ evaluated at $z=N$ has trivial reduction modulo $p$. An interesting consequence of our construction is that all polynomials $P_K(a,N)$ (mod~$p$) with bounded $a$-span are realised by knots with bounded braid index. As an application, we classify all polynomials of the form $P_K(a,1)$ (mod $2$) with $a$-span $\leq 10$.
We derive an upper bound on the density of Jones polynomials of knots modulo a prime number $p$, within a sufficiently large degree range: $4/p^7$. As an application, we classify knot Jones polynomials modulo two of span up to eight.
We derive a lower bound for all Levine-Tristram signatures of positive braid links, linear in terms of the first Betti number. As a consequence, we obtain upper and lower bounds on the ratio of fixed pairs of Levine-Tristram signature invariants, valid uniformly on all positive braid monoids.
We show that the links associated with positive elements of the Thompson group $F$ coincide with the closures of bipartite arborescent tangles.
The set of quasipositive surfaces is closed under incompressible inclusion. We prove that the induced order on fibre surfaces of positive braid links is almost a well-quasi-order. When restricting to quasipositive surfaces containing a fixed root of a full twist, we get an actual well-quasi-order.