A slice knot is a type of mathematical knot. In knot theory, a "knot" means an embedded circle in the 3sphere
and that the 3sphere can be thought of as the boundary of the fourdimensional ball
A knot is slice if it bounds a nicely embedded disk D in the 4ball.^{[1]}
What is meant by "nicely embedded" depends on the context, and there are different terms for different kinds of slice knots. If D is smoothly embedded in B^{4}, then K is said to be smoothly slice. If D is only locally flat (which is weaker), then K is said to be topologically slice.
Every ribbon knot is smoothly slice. An old question of Fox asks whether every slice knot is actually a ribbon knot.^{[2]}
The signature of a slice knot is zero.^{[3]}
The Alexander polynomial of a slice knot factors as a product where is some integral Laurent polynomial.^{[3]} This is known as the Fox–Milnor condition.^{[4]}
The following is a list of all slice knots with 10 or fewer crossings; it was compiled using the Knot Atlas^{[full citation needed]}: 6_{1},^{[5]} , , , , , , , , , , , , , , , , , , and .
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Trefoil knot
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See also
References
 ^ Lickorish, W. B. Raymond (1997), An Introduction to Knot Theory, Graduate Texts in Mathematics, 175, Springer, p. 86, ISBN 9780387982540.
 ^ Gompf, Robert E.; Scharlemann, Martin; Thompson, Abigail (2010), "Fibered knots and potential counterexamples to the property 2R and sliceribbon conjectures", Geometry & Topology, 14 (4): 2305–2347, arXiv:1103.1601, doi:10.2140/gt.2010.14.2305, MR 2740649.
 ^ ^{a} ^{b} Lickorish (1997), p. 90.
 ^ Banagl, Markus; Vogel, Denis (2010), The Mathematics of Knots: Theory and Application, Contributions in Mathematical and Computational Sciences, 1, Springer, p. 61, ISBN 9783642156373.
 ^ "6 1", The Knot Atlas.