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Autori principali:	Bellettini, Giovanni, Paolini, Giovanni, Paolini, Maurizio, Wang, Yi-Sheng
Natura:	Preprint
Pubblicazione:	2025
Soggetti:	Geometric Topology 57K12, 05C10
Accesso online:	https://arxiv.org/abs/2511.07796
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author	Bellettini, Giovanni Paolini, Giovanni Paolini, Maurizio Wang, Yi-Sheng
author_facet	Bellettini, Giovanni Paolini, Giovanni Paolini, Maurizio Wang, Yi-Sheng
contents	We study tangle replacement in the context of spatial graphs. The main results show that, for certain spatial handcuff graphs, there is a one-to-one correspondence between the neighborhood equivalence classes of the spatial graphs obtained by tangle replacement and the tangles with which the replacement is performed, up to possibly some permutation. As corollaries, we distinguish handlebody-knots difficult to differentiate with computational invariants and determine their chirality and symmetry groups.
format	Preprint
id	arxiv_https___arxiv_org_abs_2511_07796
institution	arXiv
publishDate	2025
record_format	arxiv
spellingShingle	Tangle replacement on spatial graphs Bellettini, Giovanni Paolini, Giovanni Paolini, Maurizio Wang, Yi-Sheng Geometric Topology 57K12, 05C10 We study tangle replacement in the context of spatial graphs. The main results show that, for certain spatial handcuff graphs, there is a one-to-one correspondence between the neighborhood equivalence classes of the spatial graphs obtained by tangle replacement and the tangles with which the replacement is performed, up to possibly some permutation. As corollaries, we distinguish handlebody-knots difficult to differentiate with computational invariants and determine their chirality and symmetry groups.
title	Tangle replacement on spatial graphs
topic	Geometric Topology 57K12, 05C10
url	https://arxiv.org/abs/2511.07796

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// Anecdotal testing Today and Yesterday seems to indicate destroy() // is called by the garbage collector and everything is good. // Something to keep in mind though. return 0; } /** * Write function that is called when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ public function write($sessId, $data): bool { if ($this->writesDisabled) { return true; } return $this->saveSession($sessId, $data); } /** * A function that is called internally when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ abstract protected function saveSession($sessId, $data): bool; }

* * @return void */ public function writeClose() { // The assumption is that we're using PHP's ext/session. // session_write_close() will actually overwrite $_SESSION with an // empty array on completion -- which leads to a mismatch between what // is in the storage object and $_SESSION. To get around this, we // temporarily reset $_SESSION to an array, and then re-link it to // the storage object. // // Additionally, while you _can_ write to $_SESSION following a // session_write_close() operation, no changes made to it will be // flushed to the session handler. As such, we now mark the storage // object isImmutable. $storage = $this->getStorage(); if (! $storage->isImmutable()) { $_SESSION = $storage->toArray(true); session_write_close(); $storage->fromArray($_SESSION); $storage->markImmutable(); } } /** * Attempt to set the session name * * If the session has already been started, or if the name provided fails * validation, an exception will be raised. * * @param string $name * @return SessionManager * @throws Exception\InvalidArgumentException */ public function setName($name) { if ($this->sessionExists()) { throw new Exception\InvalidArgumentException( 'Cannot set session name after a session has already started'

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