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A class of knots with simple SU(2)-representations
We call a knot in the 3-sphere SU(2)-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in SU(2) are binary dihedral. This is aExpand
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Knot concordances and alternating knots
There is an infinitely generated free subgroup of the smooth knot concordance group with the property that no nontrivial element in this subgroup can be represented by an alternating knot. ThisExpand
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Representation spaces of pretzel knots
We study the representation spaces $R(K;\bf{i})$ as appearing in Kronheimer and Mrowka's framed instanton knot Floer homology, for a class of pretzel knots. In particular, for pretzel knotsExpand
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The quantum sl(n) graph invariant and a moduli space
We associate a moduli problem to a colored trivalent graph; such graphs, when planar, appear in the state-sum description of the quantum sl(N) knot polynomial due to Murakami, Ohtsuki, and Yamada. WeExpand
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Integer homology $3$-spheres admit irreducible representations in $\operatorname{SL}(2,\mathbb{C})$
We prove that the fundamental group of any integer homology 3-sphere different from the 3-sphere admits irreducible representations of its fundamental group in SL(2,C). For hyperbolic integerExpand
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Khovanov width and dealternation number of positive braid links.
We give asymptotically sharp upper bounds for the Khovanov width and the dealternation number of positive braid links, in terms of their crossing number. The same braid-theoretic technique, combinedExpand
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Alternating numbers of torus knots with small braid index
We calculate the alternating number of torus knots with braid index 4 and less. For the lower bound, we use the upsilon-invariant recently introduced by Ozsv\'ath, Stipsicz, and Szab\'o. For theExpand
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ON LOGARITHMIC TRANSFORMATIONS ON THE HOPF SURFACE
In this note we study logarithmic transformations in the sense of differential topology on two fibers of the Hopf surface. It is known that such transformations are susceptible to yield exotic smoothExpand
Toroidal homology spheres and SU(2)-representations
We prove that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible SU(2)representations. Our methods use instanton FloerExpand
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$PU(N)$ monopoles, higher rank instantons, and the monopole invariants
A famous conjecture in gauge theory mathematics, attributed to Witten, suggests that the polynomial invariants of Donaldson are expressible in terms of the Seiberg-Witten invariants if the underlyingExpand
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