arXiv · 0708.2162
This paper has been withdrawn by the author, due to a significant error in section 4.3.1.
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Chan-Ho Suh. 2009-01-26. Normal Surface Theory in Link Diagrams. https://arxiv.org/abs/0708.2162
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We complete the work started in previous work of the author and Kalfagianni and Sikora, and give a complete description of the $\mathrm{SL}_2(\mathbb{C})$-character scheme $\mathcal{X}(M)$ of all small Seifert $3$-manifolds $M$. We find that $\mathcal{X}(M)$ is reduced if and only if $M$ admits no exceptional abelian character, and that exceptional abelian character have multiplicity $2$ in $\mathcal{X}(M).$
We derive new formulas for the Jones polynomial and the Kauffman bracket polynomial of a rational link represented by a standard diagram that is not necessarily alternating. These formulas generalize the results of Qazaqzeh, Yasein, and Abu-Qamar for the Tutte polynomial of the Tait graph of an alternating diagram of a rational link, as well as the matrix formulas of Lawrence and Rosenstein for the Jones polynomial of a rational link. Our approach uses the colored version of Brylawski's tensor product formula for Tutte polynomials of colored graphs, due to Diao, Hetyei, and Hinson. Furthermore, generalizing the formulas of Qazaqzeh, Yasein, and Abu-Qamar, we present a finite automaton that computes the crossing signs, thereby enabling the calculation of the writhe of a standard diagram of a rational link.
For a surface with an odd number of punctures, the moduli space of flat $SU(2)$ connections with traceless holonomy around each puncture is a symplectic manifold. When the moduli space is nonempty, there is a natural homomorphism from the mapping class group of the punctured surface to the symplectic mapping class group of this moduli space. It is shown that this homomorphism is injective if and only if the dimension of the moduli space is greater than $2$. This generalizes work of Seidel and Wehrheim--Woodward. Also given is a complete classification of Lagrangian spheres in the projective plane blown up at $5$ points with its monotone symplectic structure, which is the moduli space for the 5-punctured sphere. Furthermore, it is determined when two such Lagrangian spheres can be displaced by a symplectic isotopy. Results are also obtained regarding Lagrangian spheres in the intersection of two quadrics in $\mathbb{C}\mathbb{P}^5$. The proofs involve instanton Floer theory and results on Heegaard splittings. A main technical result establishes the approximation of any Hamiltonian isotopy of the $SU(2)$ moduli space by holonomy perturbations which are used in instanton homology.