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Dissertation
On the Linking Skein of a 3-manifold: Derived 3-manifold Invariants
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2020
DOI: 10.17077/etd.005491
Abstract
A topological invariant is the assignment of a mathematical object to every space so that any two homeomorphic spaces are assigned the same object. In practice, an invariant is used to distinguish topological spaces. If two spaces are assigned different objects then the two spaces can not be homeomorphic. In this thesis, we study skein modules: invariants that assign to each 3-dimensional space $M$ an algebraic module $S(M)$ which encodes how circles in $M$ can be knotted.
If $M$ is a closed 3-manifold then it admits a decomposition into two simpler pieces $M=X\cup_{\Sigma} Y$, called a Heegaard splitting. Each piece $X$ and $Y$ is a thickened planar surface, i.e. a handlebody, with common boundary surface $\Sigma$. In such a case, the associated skein module can be decomposed algebraically as $S(M)=S(X)\bigotimes_{S(\Sigma)} S(Y)$. This idea in reverse can be interpreted as combining $S(X)$ and $S(Y)$ algebraically and obtaining an invariant of $M$. Thus a natural question is: Are there other ways to combine $S(X)$ and $S(Y)$ to obtain invariants for $M$?
The Tor-modules $Tor^{S(\Sigma)}_i(S(X),S(Y))$ are the derived functors of the tensor product $S(M)=S(X)\bigotimes_{S(\Sigma)} S(Y)$, and since $Tor^{S(\Sigma)}_0(S(X),S(Y))$ is isomorphic to \\ $S(M)=S(X)\bigotimes_{S(\Sigma)} S(Y)$ the Tor-modules can be thought of as a generalization of the tensor product. Hence a tractable way to start answering the question above is to instead ask: Are the Tor-modules $Tor^{S(\Sigma)}_i(S(X),S(Y))$ invariants of $M$? In fact, the question of whether the higher Tor-modules are invariants of $M$ was raised by V. Turaev in a discussion with C. Frohman circa 2005.
In this thesis, we answer this question in the affirmative for $S(M)$ the linking skein of $M$. Specifically, we construct a chain complex whose $i$th homology is the module\\
$Tor^{S(\Sigma)}_i(S(X),S(Y))$. We then show that any two Heegaard splittings $M=X\cup_{\Sigma} Y$ and $M=X'\cup_{\Sigma'} Y'$ give rise to homotopic chain complex and thus the modules $Tor^{S(\Sigma)}_i(S(X),S(Y))$ and $Tor^{S(\Sigma')}_i(S(X'),S(Y'))$ are isomorphic. Hence the Tor-modules are invariants of $M$. Furthermore, we show that the linking skein is an abelianization related to the more complex HOMFLY-PT skein module via the concept of quantization. Therefore our result above provides evidence to support invariance of the Tor-modules in the case of the HOMFLY-PT skein.
Details
- Title: Subtitle
- On the Linking Skein of a 3-manifold: Derived 3-manifold Invariants
- Creators
- Mitchell Messmore
- Contributors
- Benjamin Cooper (Advisor)Charles Frohman (Committee Member)Keiko Kawamuro (Committee Member)Ryan Kinser (Committee Member)Miodrag Iovanov (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2020
- Publisher
- University of Iowa
- DOI
- 10.17077/etd.005491
- Number of pages
- vii, 82 pages
- Copyright
- Copyright 2020 Mitchell Messmore
- Language
- English
- Description illustrations
- illustrations (some color)
- Description bibliographic
- Includes bibliographical references (page 82).
- Academic Unit
- Mathematics
- Record Identifier
- 9983949690602771
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