arXiv subjects
Publications and source records attributed to Puttipong Pongtanapaisan.
In this paper we define the ascending number and bridge number for virtual and welded spatial graphs. We prove that the former admits a lower bound in terms of the latter. This relationship allows us to compare different notions of unknotting virtual spatial graphs by crossing changes.
We study bridge indices of graphs, virtual graphs, and their handlebody classes. We construct virtual ravels for which the difference between bridge index and weighted overpass bridge index is unbounded. The lower bound uses exponentially many colorings for one fixed flow in a $\Z_2$-family of biquandles. For each negative Euler characteristic we construct classical almost unknotted graphs of one fixed trivalent graph with arbitrarily large bridge index. We also obtain unbounded even bridge indices for ravels of each fixed bouquet rank. A separate almost unknotted family has an unbounded difference between its spatial-graph bridge index and the bridge index of its regular neighborhood. Finally, edge connected sums of Suzuki curves give exact constrained and unconstrained bridge indices whose difference is unbounded.
There are many commonly known definitions of the bridge index coming from combinatorial knot theory, Morse theory, geometry, and algebra. In this paper, we prove that the danceability index is a new equivalent definition of the bridge index which can be seen as an oriented version of the Wirtinger number. We extend the danceability invariant to virtual knots in multiple ways and compare these invariants to two different notions of bridge index for virtual knots.
We define a pants distance for knotted surfaces in 4-manifolds, which generalizes the complexity studied by Blair-Campisi-Taylor-Tomova for surfaces in the 4-sphere. We determine that if the distance computed on a given diagram does not surpass a theoretical bound in terms of the multisection genus, then the pair (X, F) admits a standard form (i.e., has simple topology). Furthermore, we calculate the exact values of our invariants for many new examples, such as the spun lens spaces. We provide a characterization of genus two quadrisections with distance at most six.
We study the entanglement complexity of a system consisting of two simple-closed curves (self-avoiding polygons) that span a lattice tube, referred to as a 2SAP. 2SAPs are of interest as the first known model of confined ring polymers where the linking probability goes to 1 exponentially with the size of the system. Atapour et al proved this in 2010 by showing that all but exponentially few sufficiently large 2SAPs contain a pattern that guarantees the 2SAP is non-split, provided that the requisite pattern fits in the tube. This result was recently extended to all tubes sizes that admit non-trivial links. Here we develop and apply knot theory results to answer more general questions about the entanglement complexity of 2SAPs. We first extend the 1992 concept of a good measure of knot complexity to a good measure, $F$, of spanning-link complexity for $k$-component links. Using tangle products, we show, for example, that the more complex the prime knot decomposition of any component of a given link type, the greater its $F$-measure. We then prove that all but exponentially few size $m$ 2SAPs have $F$ complexity that grows at least linearly in $m$ as $m\to \infty$. We establish that good measures of knot complexity yield good measures of spanning-link complexity. We also establish conditions whereby more general link invariants can yield good measures. In particular, we establish that measures based on several classical invariants are good measures by our definition, eg bridge number or the splitting number. Finally, we consider how the tube dimensions affect which links are embeddable as 2SAPs as well as geometric restrictions on the entanglement complexity of the embeddings. For example, we establish that there are two-component links that occur as 2SAPs in a given tube size only when one of the components is forced into a non-minimal bridge number conformation.
We introduce and study bridge decompositions for 3-manifolds embedded in the 5-sphere. These generalize both the classical notion of bridge position for knots in the 3-sphere and the bridge trisections of surfaces in the 4-sphere due to Meier and Zupan. Our main technical tool is the multisections of 5-manifolds introduced by Aribi, Courte, Golla, and Moussard. We prove that every embedded 3-manifold admits such a decomposition; in particular, any such embedding is encoded by four trivial tangle diagrams. We also present a range of explicit examples, including $S^2$-spun knots and ribbon 3-knots.
New explicit procedures for passing among triplane diagrams, braid movies, and braid charts for knotted surfaces in $\mathbb{R}^4$ are presented. To this end, rainbow diagrams, which lie between braid charts and triplanes, are introduced. Inequalities relating the braid index and the bridge index of 2-knots are obtained via these procedures. Another consequence is a 4-dimensional version of the classical result that ``the minimal number of Seifert circles equals the braid index of a link'' due to Yamada. The procedures are exemplified for the spun trefoil, the 2-twist spun trefoil, and other related examples. Of independent interest, an appendix is included that describes a procedure for drawing a triplane diagram for a satellite surface with a 2-sphere companion. Thus, larger families of surfaces for which we know specific triplane diagrams are obtained.
We extend the Wirtinger number of links, an invariant originally defined by Blair, Kjuchukova, Velazquez, and Villanueva in terms of extending initial colorings of some strands of a diagram to the entire diagram, to spatial graphs. We prove that the Wirtinger number equals the bridge index of spatial graphs, and we implement an algorithm in Python which gives a more efficient way to estimate upper bounds of bridge indices. Combined with lower bounds from diagram colorings by elements from certain algebraic structures and clasping techniques, we obtain exact bridge indices for a large family of almost unknotted spatial graphs. We also show that for every possible negative Euler characteristic, there exist almost unknotted graphs of arbitrarily large bridge index.
We study distance relations in various simplicial complexes associated with low-dimensional manifolds. In particular, complexes satisfying certain topological conditions with vertices as simple multi-curves. We obtain bounds on the distances in such complexes in terms of number of components in the vertices and distance in the curve complex. We then define new invariants for closed 3-manifolds and handlebody-knots. These are defined using the splitting distance which is calculated using the distance in a simplicial complex associated with the splitting surface arising from the Heegard decompositions of the 3-manifold. We prove that the splitting distances in each case is bounded from below under stabilizations and as a result the associated invariants converge to a non-trivial limit under stabilizations.
We investigate connections between biquandle colorings, quiver enhancements, and several notions of the bridge numbers $b_i(K)$ for virtual links, where $i=1,2$. We show that for any positive integers $m \leq n$, there exists a virtual link $K$ with $b_1(K) = m$ and $b_2(K) = n$, thereby answering a question posed by Nakanishi and Satoh. In some sense, this gap between the two formulations measures how far the knot is from being classical. We also use these bridge number analyses to systematically construct families of links in which quiver invariants can distinguish between links that share the same biquandle counting invariant.
For any pair of integers $m$ and $n$ such that $3<m<n$, we provide an infinite family of links, where each link in the family has a locally minimal $n$-bridge position and a globally minimal $m$-bridge position. We accomplish this by applying the criterion of Takao et al. The $n$-bridge position is interesting because the corresponding bridge sphere is unperturbed, so it must be perturbed at least once before it can be de-perturbed to attain a globally minimal $m$-bridge sphere.
This paper employs various computational techniques to determine the bridge numbers of both classical and virtual knots. For classical knots, there is no ambiguity of what the bridge number means. For virtual knots, there are multiple natural definitions of bridge number, and we demonstrate that the difference can be arbitrarily far apart. We then acquired two datasets, one for classical and one for virtual knots, each comprising over one million labeled data points. With the data, we conduct experiments to evaluate the effectiveness of common machine learning models in classifying knots based on their bridge numbers.
In this paper, we define invariants of links in terms of colorings of link diagrams and prove that these invariants coincide with various notions of widths of links with respect to the standard Morse function. Our formulations are advantageous because they are algorithmic and suitable for program implementations. As an application, we calculate the max-width of over 10000 links up to 14 crossings from the link table.
Two isomorphic graphs can have inequivalent spatial embeddings in 3-space. In this way, an isomorphism class of graphs contains many spatial graph types. A common way to measure the complexity of a spatial graph type is to count the minimum number of straight sticks needed for its construction in 3-space. In this paper, we give estimates of this quantity by enumerating stick diagrams in a plane. In particular, we compute the planar stick indices of knotted graphs with low crossing numbers. We also show that if a bouquet graph or a theta-curve has the property that its proper subgraphs are all trivial, then the planar stick index must be at least seven.
The quandle coloring quiver was introduced by Cho and Nelson as a categorification of the quandle coloring number. In some cases, it has been shown that the quiver invariant offers more information than other quandle enhancements. In this paper, we compute the quandle coloring quivers of 2-bridge links with respect to the dihedral quandles.
A bridge sphere is said to be keen weakly reducible if it admits a unique pair of disjoint compressing disks on opposite sides. In particular, such a bridge sphere is weakly reducible, not perturbed, and not topologically minimal in the sense of David Bachman. In terms of Jennifer Schultens' width complex, a link in bridge position with respect to a keen weakly reducible bridge sphere is distance one away from a local minimum. In this paper, we give infinitely many examples of keen weakly reducible bridge spheres for links in $b$ bridge position for $b \geq 4.$
The Meridional Rank Conjecture asks whether the bridge number of a knot in $S^3$ is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper we investigate the analogous conjecture for knotted surfaces in $S^4$. Towards this end, we give a construction to produce classical knots with quotients sending meridians to elements of any finite order and which detect their meridional ranks. We establish the equality of bridge number and meridional rank for these knots and knotted spheres obtained from them by twist-spinning. On the other hand, we show that the meridional rank of knotted spheres is not additive under connected sum, so that either bridge number also collapses, or meridional rank is not equal to bridge number for knotted spheres. We also show a relationship between the bridge numbers of welded knots and ribbon tori using the Tube map, and give applications to bridge trisections of knotted surfaces.
Although rare, an increasing number of proteins have been observed to contain entanglements in their native structures. To gain more insight into the significance of protein knotting, researchers have been investigating protein knot formation using both experimental and theoretical methods. Motivated by the hypothesized folding pathway of $α$-haloacid dehalogenase (DehI) protein, Flapan, He, and Wong proposed a theory of how protein knots form, which includes existing folding pathways described by Taylor and Bölinger et al. as special cases. In their topological descriptions, two loops in an unknotted open protein chain containing at most two twists each come close together, and one end of the protein eventually passes through the two loops. In this paper, we build on Flapan, He, and Wong's theory where we pay attention to the crossing signs of the threading process and assume that the unknotted protein chain may arrange itself into a more complicated configuration before threading occurs. We then apply tangle calculus, originally developed by Ernst and Sumners to analyze the action of specific proteins on DNA, to give all possible knots or knotoids that may be discovered in the future according to our model and give recipes for engineering specific knots in proteins from simpler pieces. We show why twists knots are the most likely knots to occur in proteins. We use chirality to show that the most likely knots to occur in proteins via Taylor's twisted hairpin model are the knots $+3_1$, $4_1$, and $-5_2$.