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You searched for subject:(Periodic Knots). Showing records 1 – 3 of 3 total matches.

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McMaster University

1. White, Lindsay. Alexander Invariants of Periodic Virtual Knots.

Degree: PhD, 2017, McMaster University

In this thesis, we show that every periodic virtual knot can be realized as the closure of a periodic virtual braid. If K is a q-periodic virtual knot with quotient K_*, then the knot group GK_* is a quotient of GK and we derive an explicit q-symmetric Wirtinger presentation for GK, whose quotient is a Wirtinger presentation for GK_*. When K is an almost classical knot and q=pr, a prime power, we show that K_* is also almost classical, and we establish a Murasugi-like congruence relating their Alexander polynomials modulo p. This result is applied to the problem of determining the possible periods of a virtual knot K. For example, if K is an almost classical knot with nontrivial Alexander polynomial, our result shows that K can be p-periodic for only finitely many primes p. Using parity and Manturov projection, we are able to apply the result and derive conditions that a general q-periodic virtual knot must satisfy. The thesis includes a table of almost classical knots up to 6 crossings, their Alexander polynomials, and all known and excluded periods.

Thesis

Doctor of Philosophy (PhD)

Advisors/Committee Members: Boden, Hans U., Nicas, Andrew J., Mathematics.

Subjects/Keywords: Knot Theory; Virtual Knots; Periodic Knots; Virtual Knot Theory

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APA (6th Edition):

White, L. (2017). Alexander Invariants of Periodic Virtual Knots. (Doctoral Dissertation). McMaster University. Retrieved from http://hdl.handle.net/11375/21006

Chicago Manual of Style (16th Edition):

White, Lindsay. “Alexander Invariants of Periodic Virtual Knots.” 2017. Doctoral Dissertation, McMaster University. Accessed August 13, 2020. http://hdl.handle.net/11375/21006.

MLA Handbook (7th Edition):

White, Lindsay. “Alexander Invariants of Periodic Virtual Knots.” 2017. Web. 13 Aug 2020.

Vancouver:

White L. Alexander Invariants of Periodic Virtual Knots. [Internet] [Doctoral dissertation]. McMaster University; 2017. [cited 2020 Aug 13]. Available from: http://hdl.handle.net/11375/21006.

Council of Science Editors:

White L. Alexander Invariants of Periodic Virtual Knots. [Doctoral Dissertation]. McMaster University; 2017. Available from: http://hdl.handle.net/11375/21006


Uniwersytet im. Adama Mickiewicza w Poznaniu

2. Politarczyk, Wojciech. Homologie Khovanova splotów symetrycznych .

Degree: 2015, Uniwersytet im. Adama Mickiewicza w Poznaniu

Rozprawa ta prezentuje konstrukcję wariantu homologii Khovanova dla tzw. splotów periodycznych, czyli splotów posiadających pewną symetrię. Ta wersja homologii Khovanova uwzględnia symetrie splotów. Przy pomocy metod algebry homologicznej, takich jak funktory pochodne i ciągi spektralne, oraz teorii całkowitoliczbowych reprezentacji grup cyklicznych podajemy konstrukcję i opisujemy podstawowe własności ekwiwariantnych homologii Khovanova. Dodatkowo, konstruujemy ciąg spektralny, który pozwala wyliczać ekwiwariantne homologie Khovanova. Ciąg ten jest adaptacją motkowego ciągu dokładnego. W dalszej części wyliczany wymierne ekwiwariantne homologie Khovanova splotów torusowych T(n,2). Oprócz tego, rozważamy ekwiwariantne odpowiedniki wielomianu Jonesa. Pokazujemy, że spełniają one odpowiednik relacji motkowej dla klasycznego wielomianu Jonesa i używamy tej własności do wzmocnienia kryterium periodyczności splotu podanego przez J.H. Przytyckiego. Dodatkowo, wyprowadzamy sumę statystyczną dla ekwiwariantnych odpowiedników wielomianu Jonesa. Konsekwencją tego faktu jest klasyczna kongruencja podana przez K. Murasugiego. Advisors/Committee Members: Pawałowski, Krzysztof. Promotor (advisor).

Subjects/Keywords: periodyczne węzły; periodic knots; periodyczne sploty; periodic links; homologie Khovanova; Khovanov homology; wielomian Jonesa; Jones polynomial

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Politarczyk, W. (2015). Homologie Khovanova splotów symetrycznych . (Doctoral Dissertation). Uniwersytet im. Adama Mickiewicza w Poznaniu. Retrieved from http://hdl.handle.net/10593/13034

Chicago Manual of Style (16th Edition):

Politarczyk, Wojciech. “Homologie Khovanova splotów symetrycznych .” 2015. Doctoral Dissertation, Uniwersytet im. Adama Mickiewicza w Poznaniu. Accessed August 13, 2020. http://hdl.handle.net/10593/13034.

MLA Handbook (7th Edition):

Politarczyk, Wojciech. “Homologie Khovanova splotów symetrycznych .” 2015. Web. 13 Aug 2020.

Vancouver:

Politarczyk W. Homologie Khovanova splotów symetrycznych . [Internet] [Doctoral dissertation]. Uniwersytet im. Adama Mickiewicza w Poznaniu; 2015. [cited 2020 Aug 13]. Available from: http://hdl.handle.net/10593/13034.

Council of Science Editors:

Politarczyk W. Homologie Khovanova splotów symetrycznych . [Doctoral Dissertation]. Uniwersytet im. Adama Mickiewicza w Poznaniu; 2015. Available from: http://hdl.handle.net/10593/13034


Australian National University

3. Evans, Myfanwy Ella. Three-dimensional entanglement: knots, knits and nets .

Degree: 2011, Australian National University

Three-dimensional entanglement, including knots, periodic arrays of woven filaments (weavings) and periodic arrays of interpenetrating networks (nets), forms an integral part of the analysis of structure within the natural sciences. This thesis constructs a catalogue of 3-periodic entanglements via a scaffold of Triply-Periodic Minimal Surfaces (TPMS). The two-dimensional Hyperbolic plane can be wrapped over a TPMS in much the same way as the two-dimensional Euclidean plane can be wrapped over a cylinder. Thus vertices and edges of free tilings of the Hyperbolic plane, which are tilings by tiles of infinite size, can be wrapped over a TPMS to represent vertices and edges of an array in three-dimensional Euclidean space. In doing this, we harness the simplicity of a two-dimensional surface as compared with 3D space to build our catalogue. We numerically tighten these entangled flexible knits and nets to an ideal conformation that minimises the ratio of edge (or filament) length to diameter. To enable the tightening of periodic entanglements which may contain vertices, we extend the Shrink-On-No-Overlaps algorithm, a simple and fast algorithm for tightening finite knots and links. The ideal geometry of 3-periodic weavings found through the tightening process exposes an interesting physical property: Dilatancy. The cooperative straightening of the filaments with a fixed diameter induces an expansion of the material accompanied with an increase in the free volume of the material. Further, we predict a dilatant rod packing as the structure of the keratin matrix in the corneocytes of mammalian skin, where the dilatant property of the matrix allows the skin to maintain structural integrity while experiencing a large expansion during the uptake of water.

Subjects/Keywords: filament packings; entanglement; triply-periodic minimal surfaces; hyperbolic tilings; entangled networks; rod packings; ideal knots; stratum corneum swelling

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Evans, M. E. (2011). Three-dimensional entanglement: knots, knits and nets . (Thesis). Australian National University. Retrieved from http://hdl.handle.net/1885/9502

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Evans, Myfanwy Ella. “Three-dimensional entanglement: knots, knits and nets .” 2011. Thesis, Australian National University. Accessed August 13, 2020. http://hdl.handle.net/1885/9502.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Evans, Myfanwy Ella. “Three-dimensional entanglement: knots, knits and nets .” 2011. Web. 13 Aug 2020.

Vancouver:

Evans ME. Three-dimensional entanglement: knots, knits and nets . [Internet] [Thesis]. Australian National University; 2011. [cited 2020 Aug 13]. Available from: http://hdl.handle.net/1885/9502.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Evans ME. Three-dimensional entanglement: knots, knits and nets . [Thesis]. Australian National University; 2011. Available from: http://hdl.handle.net/1885/9502

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

.