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BorlandThesis.pdf (901.71 KB)
ETD Abstract Container
Abstract Header
An Invariant of Links on Surfaces via Hopf Algebra Bundles
Author Info
Borland, Alexander I
Permalink:
http://rave.ohiolink.edu/etdc/view?acc_num=osu1503183775028923
Abstract Details
Year and Degree
2017, Doctor of Philosophy, Ohio State University, Mathematics.
Abstract
One semi-classical knot invariant involves turning a knot diagram into a curve in ℝ
2
which is "decorated" by elements of a ribbon Hopf algebra H. A decorated curve is turned into an element of H using a form of pictoral calculus. The image of this element in a certain quotient space of H defines a framed knot invariant. We generalize this process to define an invariant of links in a thickened surface Σ × [0, 1], where Σ is a connected, oriented surface. In this process, we develop a theory of decorated curves in an arbitrary smooth manifold M using a balanced, flat ribbon Hopf algebra bundle E → M with typical fiber H. The link invariant is defined using decorated curves in the unit tangent bundle of T
1
Σ, and takes values in the quotient space of the semi-direct product k[ π
1
(M , b_0) ] ⊗ H. We also define local diagrams to picture the decorated curves. The original pictoral calculus for decorated curves in ℝ
2
is recaptured by viewing decorated curves in T
1
Σ through these local diagrams.
Committee
Thomas Kerler (Advisor)
Sergei Chmutov (Committee Member)
Henri Moscovici (Committee Member)
Pages
208 p.
Subject Headings
Mathematics
Keywords
Hopf Algebra
;
Quantum Group
;
Knot Invariant
;
Link Invariant
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Citations
Borland, A. I. (2017).
An Invariant of Links on Surfaces via Hopf Algebra Bundles
[Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1503183775028923
APA Style (7th edition)
Borland, Alexander.
An Invariant of Links on Surfaces via Hopf Algebra Bundles.
2017. Ohio State University, Doctoral dissertation.
OhioLINK Electronic Theses and Dissertations Center
, http://rave.ohiolink.edu/etdc/view?acc_num=osu1503183775028923.
MLA Style (8th edition)
Borland, Alexander. "An Invariant of Links on Surfaces via Hopf Algebra Bundles." Doctoral dissertation, Ohio State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=osu1503183775028923
Chicago Manual of Style (17th edition)
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Document number:
osu1503183775028923
Download Count:
300
Copyright Info
© 2017, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.