An analog to intrinsic linking, intrinsic even linking, is explored in the first half of this paper. Four graphs are established to be minor minimal intrinsically even linked, and it is conjectured that they form a complete minor minimal set. Some characterizations are given, using the simplest of the four graphs as an integral part of the arguments, that may be useful in proving the conjecture. The second half of this paper investigates a new approach to intrinsic knotting. By adapting knot energy to graphs, it is hoped that intrinsic knotting can be detected through direct computation. However, graph energies are difficult to compute, and it is unclear whether they can be used to determine whether a graph is intrinsically knotted.

This report combines a series of papers to trace progression in the area of type-2 topoisomerase research. First, Deibler et al. show that knotted DNA is harmful to cells. Knots can block both transcription and replication, and can also act as a catalyst for mutation. Despite the fact that type-2 topoisomerases perform the important functions of unknotting and decatenating DNA, the mechanism by which they accomplish this task is still unknown. Buck and Zechiedrich propose a model in which the enzyme uses local geometry to infer global topology, and thus where to perform segment passage in order to obtain the desired results. In two articles, Liu et al. evaluate this theory quantitatively for the decatenation and unknotting problems. In both cases it is shown that the presence of certain juxtapositions is strongly correlated with global topology. This correlation is not enough, however, and Liu et al. go on to show that when segment passage operations designed to mimic type-2 topoisomerase action are performed at hooked juxtapositions, the overwhelming tendency is towards unknotting and decatenation. / text

We study here a model for a strand passage in a ring polymer about a randomly chosen location at which two strands of the polymer have been brought gcloseh together. The model is based on ¦-SAPs, which are unknotted self-avoiding polygons in Z^3 that contain a fixed structure ¦ that forces two segments of the polygon to be close together. To study this model, the Composite Markov Chain Monte Carlo (CMCMC) algorithm, referred to as the CMC ¦-BFACF algorithm, that I developed and proved to be ergodic for unknotted ¦-SAPs in my M. Sc. Thesis, is used. Ten simulations (each consisting of 9.6~10^10 time steps) of the CMC ¦-BFACF algorithm are performed and the results from a statistical analysis of the simulated data are presented. To this end, a new maximum likelihood method, based on previous work of Berretti and Sokal, is developed for obtaining maximum likelihood estimates of the growth constants and critical exponents associated respectively with the numbers of unknotted (2n)-edge ¦-SAPs, unknotted (2n)-edge successful-strand-passage ¦-SAPs, unknotted (2n)-edge failed-strand-passage ¦-SAPs, and (2n)-edge after-strand-passage-knot-type-K unknotted successful-strand-passage ¦-SAPs. The maximum likelihood estimates are consistent with the result (proved here) that the growth constants are all equal, and provide evidence that the associated critical exponents are all equal.<p> We then investigate the question gGiven that a successful local strand passage occurs at a random location in a (2n)-edge knot-type K ¦-SAP, with what probability will the ¦-SAP have knot-type Kf after the strand passage?h. To this end, the CMCMC data is used to obtain estimates for the probability of knotting given a (2n)-edge successful-strand-passage ¦-SAP and the probability of an after-strand-passage polygon having knot-type K given a (2n)-edge successful-strand-passage ¦-SAP. The computed estimates numerically support the unproven conjecture that these probabilities, in the n¨ limit, go to a value lying strictly between 0 and 1. We further prove here that the rate of approach to each of these limits (should the limits exist) is less than exponential.<p> We conclude with a study of whether or not there is a difference in the gsizeh of an unknotted successful-strand-passage ¦-SAP whose after-strand-passage knot-type is K when compared to the gsizeh of a ¦-SAP whose knot-type does not change after strand passage. The two measures of gsizeh used are the expected lengths of, and the expected mean-square radius of gyration of, subsets of ¦-SAPs. How these two measures of gsizeh behave as a function of a polygonfs length and its after-strand-passage knot-type is investigated.

We study here a model for a strand passage in a ring polymer about a randomly chosen location at which two strands of the polymer have been brought gcloseh together. The model is based on ¦-SAPs, which are unknotted self-avoiding polygons in Z^3 that contain a fixed structure ¦ that forces two segments of the polygon to be close together. To study this model, the Composite Markov Chain Monte Carlo (CMCMC) algorithm, referred to as the CMC ¦-BFACF algorithm, that I developed and proved to be ergodic for unknotted ¦-SAPs in my M. Sc. Thesis, is used. Ten simulations (each consisting of 9.6~10^10 time steps) of the CMC ¦-BFACF algorithm are performed and the results from a statistical analysis of the simulated data are presented. To this end, a new maximum likelihood method, based on previous work of Berretti and Sokal, is developed for obtaining maximum likelihood estimates of the growth constants and critical exponents associated respectively with the numbers of unknotted (2n)-edge ¦-SAPs, unknotted (2n)-edge successful-strand-passage ¦-SAPs, unknotted (2n)-edge failed-strand-passage ¦-SAPs, and (2n)-edge after-strand-passage-knot-type-K unknotted successful-strand-passage ¦-SAPs. The maximum likelihood estimates are consistent with the result (proved here) that the growth constants are all equal, and provide evidence that the associated critical exponents are all equal.<p> We then investigate the question gGiven that a successful local strand passage occurs at a random location in a (2n)-edge knot-type K ¦-SAP, with what probability will the ¦-SAP have knot-type Kf after the strand passage?h. To this end, the CMCMC data is used to obtain estimates for the probability of knotting given a (2n)-edge successful-strand-passage ¦-SAP and the probability of an after-strand-passage polygon having knot-type K given a (2n)-edge successful-strand-passage ¦-SAP. The computed estimates numerically support the unproven conjecture that these probabilities, in the n¨ limit, go to a value lying strictly between 0 and 1. We further prove here that the rate of approach to each of these limits (should the limits exist) is less than exponential.<p> We conclude with a study of whether or not there is a difference in the gsizeh of an unknotted successful-strand-passage ¦-SAP whose after-strand-passage knot-type is K when compared to the gsizeh of a ¦-SAP whose knot-type does not change after strand passage. The two measures of gsizeh used are the expected lengths of, and the expected mean-square radius of gyration of, subsets of ¦-SAPs. How these two measures of gsizeh behave as a function of a polygonfs length and its after-strand-passage knot-type is investigated.

In this thesis we study the problem of robotic interaction with objects from a flexible perspective that complements the rigid force-closure approach. In a flexible interaction the object is not firmly bound to the robot (immobilized), which leads to many interesting scenarios. We focus on the secure kind of flexible interactions, commonly referred to as caging grasps. In this context, the adjective secure implies that the object is not able to escape arbitrarily far away from the robot which is caging it. A cage is a secure flexible interaction because it does not immobilize the object, but restricts its motion to a finite set of possible configurations. We study cages in two novel scenarios for objects with holes: caging through multi-agent cooperation and through dual-arm knotting with a rope. From these two case studies, we were able to analyze the caging problem in a broader perspective leading to the definition of a hierarchical classification of flexible interactions and cages. In parallel to the geometric and physical problem of flexible interactions with objects, we study also the problem of discrete action scheduling through a novel control architecture called Behavior Trees (BTs). In this thesis we propose a formulation that unifies the competing BT philosophies into a single framework. We analyze how the mainstream BT formulations differ from each other, as well as their benefits and limitations. We also compare the plan representation capabilities of BTs with respect to the traditional approach of Controlled Hybrid Dynamical Systems (CHDSs). In this regard, we present bidirectional translation algorithms between such representations as well as the necessary and sufficient conditions for translation convergence. Lastly, we demonstrate our action scheduling BT architecture showcasing the aforementioned caging scenarios, as well as other examples that show how BTs can be interfaced with other high level planners. / <p>QC 20170322</p>

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