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Slicing degree of knots
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abstract
The slicing degree of a knot $K$ is defined as the smallest integer $k$ such that $K$ is $k$-slice in $\#^n \overline{\mathbb{CP}^2}$ for some $n$. In this paper, we establish bounds for the slicing degrees of knots using Rasmussen's $s$-invariant, knot Floer homology and singular instanton homology. We compute the slicing degrees for many small knots (with crossing numbers up to $9$) and for some families of torus knots.
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Simply slicing knots
A knot K bounds a locally flat Z_d-disc representing a class x of divisibility d in a simply-connected 4-manifold N iff its Arf invariant matches a congruence and b_2(N) dominates all Levine-Tristram signature bounds.
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