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Register a new userBalanced algebraic unknotting, linking forms, and surfaces in three- and four-space
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We provide three 3-dimensional characterizations of the Z-slice genus of a knot, the minimum genus of a locally-flat surface in 4-space cobounding the knot whose complement has cyclic fundamental group: in terms of balanced algebraic unknotting, in terms of Seifert surfaces, and in terms of presentation matrices of the Blanchfield pairing. This result generalizes to a knot in an integer homology 3-sphere and surfaces in certain simply connected signature zero 4-manifolds cobounding this homology sphere. Using the Blanchfield characterization, we obtain effective lower bounds for the Z-slice genus from the linking pairing of the double branched cover of the knot. In contrast, we show that for odd primes p, the linking pairing on the first homology of the p-fold branched cover is determined up to isometry by the action of the deck transformation group on said first homology.
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Two triples of triangles having pairwise disjoint outlines in 3-space are called combinatorially isotopic if one triple can be obtained from the other by a continuous motion during which the outlines of the triangles remain pairwise disjoint. We conjecture that it can be algorithmically checked if an (ordered or unordered) triple of triangles is combinatorially isotopic to a triple of triangles having pairwise disjoint convex hulls. We also conjecture that any unordered triple of pairwise disjoint triangles in 3-space belongs to one of the 5 types of such triples listed in the paper. We present an elementary proof that triples of different types are not combinatorially isotopic.
Given a knot K in S^3, let u^-(K) (respectively, u^+(K)) denote the minimum number of negative (respectively, positive) crossing changes among all unknotting sequences for K. We use knot Floer homology to construct the invariants l^-(K), l^+(K) and l(K), which give lower bounds on u^-(K), u^+(K) and the unknotting number u(K), respectively. The invariant l(K) only vanishes for the unknot, and is greater than or equal to the u^-(K). Moreover, the difference l(K)- u^-(K) can be arbitrarily large. We also present several applications towards bounding the unknotting number, the alteration number and the Gordian distance.
We prove that under some assumptions on the mean curvature the set of umbilical points of an immersed surface in a $3$-dimensional space form has positive measure. In case of an immersed sphere our result can be seen as a generalization of the celebrated Hopf theorem.
Using Seiberg-Witten Floer spectrum and Pin(2)-equivariant KO-theory, we prove new Furuta-type inequalities on the intersection forms of spin cobordisms between homology $3$-spheres. As an application, we give explicit constrains on the intersection forms of spin $4$-manifolds bounded by Brieskorn spheres $pmSigma(2,3,6kpm1)$. Along the way, we also give an alternative proof of Furuta-Kametannis improvement of 10/8-theorem for closed spin-4 manifolds.
We show there exist tunnel number one hyperbolic 3-manifolds with arbitrarily long unknotting tunnel. This provides a negative answer to an old question of Colin Adams.
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