Extending Topological Surgery to Natural Processes and Dynamical Systems

Topological surgery is a mathematical technique used for creating new manifolds out of known ones. We observe that it occurs in natural phenomena where a sphere of dimension 0 or 1 is selected, forces are applied and the manifold in which they occur changes type. For example, 1-dimensional surgery happens during chromosomal crossover, DNA recombination and when cosmic magnetic lines reconnect, while 2-dimensional surgery happens in the formation of tornadoes, in the phenomenon of Falaco solitons, in drop coalescence and in the cell mitosis. Inspired by such phenomena, we introduce new theoretical concepts which enhance topological surgery with the observed forces and dynamics. To do this, we first extend the formal definition to a continuous process caused by local forces. Next, for modeling phenomena which do not happen on arcs or surfaces but are 2-dimensional or 3-dimensional, we fill in the interior space by defining the notion of solid topological surgery. We further introduce the notion of embedded surgery in $S^3$ for modeling phenomena which involve more intrinsically the ambient space, such as the appearance of knotting in DNA and phenomena where the causes and effect of the process lies beyond the initial manifold, such as the formation of black holes. Finally, we connect these new theoretical concepts with a dynamical system and we present it as a model for both 2-dimensional 0-surgery and natural phenomena exhibiting a `hole drilling' behavior. We hope that through this study, topology and dynamics of many natural phenomena, as well as topological surgery itself, will be better understood.Comment: 54 pages, 34 figure

Eliciting Co-Creation Best Practices of Virtual Reality Reusable e-Resources

Immersive experiential technologies find fertile grounds to grow and support healthcare education. Virtual, Augmented, or Mixed reality (VR/AR/MR) have proven to be impactful in both the educational and the affective state of the healthcare student’s increasing engagement. However, there is a lack of guidance for healthcare stakeholders on developing and integrating virtual reality resources into healthcare training. Thus, the authors applied Bardach’s Eightfold Policy Analysis Framework to critically evaluate existing protocols to determine if they are inconsistent, ineffective, or result in uncertain outcomes, following systematic pathways from concepts to decision-making. Co-creative VR resource development resulted as the preferred method. Best practices for co-creating VR Reusable e-Resources identified co-creation as an effective pathway to the prolific use of immersive media in healthcare education. Co-creation should be considered in conjunction with a training framework to enhance educational quality. Iterative cycles engaging all stakeholders enhance educational quality, while co-creation is central to the quality assurance process both for technical and topical fidelity, and tailoring resources to learners’ needs. Co-creation itself is seen as a bespoke learning modality. This paper provides the first body of evidence for co-creative VR resource development as a valid and strengthening method for healthcare immersive content development. Despite prior research supporting co-creation in immersive resource development, there were no established guidelines for best practices

Μαθηματική μοντελοποίηση μέσω τοπολογικής χειρουργικής και εφαρμογές

Topological surgery is a mathematical technique used for creating new manifolds out of known ones. We observe that it occurs in natural phenomena where forces are applied and the manifold in which they occur changes type. For example, 1-dimensional surgery happens during chromosomal crossover, DNA recombination and when cosmic magnetic lines reconnect, while 2-dimensional surgery happens in the formation of Falaco solitons, in drop coalescence and in the cell mitosis. Inspired by such phenomena, we enhance topological surgery with the observed forces and dynamics. We then generalize these low-dimensional cases to a model which extends the formal definition to a continuous process caused by local forces for an arbitrary dimension m. Next, for modeling phenomena which do not happen on arcs, respectively surfaces, but are 2-dimensional, respectively 3-dimensional, we fill in the interior space by defining the notion of solid topological surgery. We further present a dynamical system as a model for both natural phenomena exhibiting a`hole drilling' behavior and our enhanced notion of solid 2-dimensional 0-surgery. Moreover, we analyze the ambient space (which we consider to be the 3-sphere) in order to introduce the notion of embedded topological surgery in the 3-sphere. This notion is then used for modeling phenomena which involve more intrinsically the ambient space, such as the appearance of knotting in DNA and phenomena where the causes and effects of the process lie beyond the initial manifold, such as the formation of tornadoes. Moreover, we present a visualization of the 4-dimensional process of 3-dimensional surgery by using the new notion of decompactified 2-dimensional surgery and rotations. Finally, we propose a model for a phenomenon exhibiting 3-dimensional surgery: the formation of black holes from cosmic strings. We hope that through this study, topology and dynamics of many natural phenomena, as well as topological surgery itself, will be better understood.Η τοπολογική χειρουργική χρησιμοποιείται για την κατασκευή νέων πολλαπλοτήτων από γνωστές. Παρατηρήσαμε ότι πολλά φαινόμενα στη φύση και πολλές φυσικές διεργασίες, τόσο του μικροκόσμου όσο και του μακροκόσμου, μπορούν να εξηγηθούν μέσω της τοπολογικής χειρουργικής σε μία, δύο και τρεις διαστάσεις. Προκειμένου να δώσουμε επιτυχή τοπολογικά μοντέλα για αυτά τα φαινόμενα, εισαγάγαμε νέες έννοιες στην τοπολογική χειρουργική όπως: εισαγωγή δυνάμεων και κέντρων έλξης, εισαγωγή της έννοιας της στερεάς χειρουργικής καθώς και της έννοιας της εμβαπτισμένης χειρουργικής στον τρισδιάστατο χώρο. Για παράδειγμα, η έννοια της εμβαπτισμένης 1-διάστατης 0-χειρουργικής επιτρέπει τη δημιουργία κόμβων και μπορεί να περιγράψει φαινόμενα όπως η αναδιάταξη του DNA, η δημιουργία νέων γονιδίων κατά την διαδικασία της μείωσης και η μαγνητική επανασύνδεση των κοσμικών μαγνητικών γραμμών. Ένα παράδειγμα στις 2 διαστάσεις είναι η έννοια της εμβαπτισμένης στερεής 2-διάστατης 0-χειρουργικής η οποία μας επιτρέπει να εξετάσουμε τί συμβαίνει και στον υπόλοιπο χώρο. Το συγκεκριμένο μοντέλο μπορεί να χρησιμοποιηθεί για να εξηγήσει τοπολογικά μη τοπικά φαινόμενα όπως είναι η δημιουργία τυφώνων αλλά και η δημιουργία των Falaco solitons. Επιπλέον, συνδέουμε την τοπολογική χειρουργική με ένα τρισδιάστατο μη-γραμμικό δυναμικό σύστημα Lotka-Volterra και αντιστοιχούμε τα ποιοτικά χαρακτηριστικά του συστήματος με τα βασικά στοιχεία της χειρουργικής. Το τελευταίο μέρος του διδακτορικού αφορά τους τρόπους απεικόνισης της τρισδιάστατης χειρουργικής και την σύνδεση αυτής με κοσμολογικά φαινόμενα. Καθώς η τρισδιάστατη χειρουργική είναι μια διαδικασία η οποία συμβαίνει στο τετρασδιάστατο χώρο, η απεικόνιση της χρειάστηκε να χρησιμοποιήσουμε τοπολογικά εργαλεία και ιδιότητες που παρατηρήσαμε στις χαμηλότερες διαστάσεις. Τέλος, δείχνουμε πώς η τρισδιάστατη χειρουργική σχετίζεται με τις μελανές οπές παρουσιάζοντας ένα καινούργιο μοντέλο για την δημιουργία των μελανών οπών από κοσμικές χορδές

Passing from (a) <i>S</i>³ as two solid tori to (b) <i>S</i>³ as two balls.

<p>Passing from (a) <i>S</i>³ as two solid tori to (b) <i>S</i>³ as two balls.</p

Soap bubble splitting.

<p>An example of 2-dimensional 1-surgery.</p