Slice knot

A slice knot is a type of mathematical knot. In knot theory, a "knot" means an embedded circle in the 3-sphere

S^{3}=\{{\mathbf {x}}\in {\mathbb {R}}^{4}\mid |{\mathbf {x}}|=1\}

and that the 3-sphere can be thought of as the boundary of the four-dimensional ball

B^{4}=\{{\mathbf {x}}\in {\mathbb {R}}^{4}\mid |{\mathbf {x}}|\leq 1\}.

A knot $K\subset S^{3}$ is slice if it bounds a nicely embedded disk D in the 4-ball.^[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 B⁴, 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 $f(t)f(t^{{-1}})$ where $f(t)$ 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: 6₁, $8_{8}$ , $8_{9}$ , $8_{{20}}$ , $9_{{27}}$ , $9_{{41}}$ , $9_{{46}}$ , $10_{3}$ , $10_{{22}}$ , $10_{{35}}$ , $10_{{42}}$ , $10_{{48}}$ , $10_{{75}}$ , $10_{{87}}$ , $10_{{99}}$ , $10_{{123}}$ , $10_{{129}}$ , $10_{{137}}$ , $10_{{140}}$ , $10_{{153}}$ and $10_{{155}}$ .

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 slice-ribbon conjectures", Geometry & Topology, 14 (4): 2305–2347, doi:10.2140/gt.2010.14.2305, MR 2740649 .
1 2 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 .

Knot theory (knots and links)

Hyperbolic	Figure-eight (4₁) Three-twist (5₂) Stevedore (6₁) 6₂ 6₃ Endless (7₄) Carrick mat (8₁₈) Perko pair (10₁₆₁) (−2,3,7) pretzel (12n242) Whitehead (52 1) Borromean rings (63 2) L10a140

Satellite	Composite knots Granny Square Knot sum

Torus	Unknot (0₁) Trefoil (3₁) Cinquefoil (5₁) Septafoil (7₁) Unlink (02 1) Hopf (22 1) Solomon's (42 1)

Invariants	Alternating Arf invariant Bridge no. 2-bridge Brunnian Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability Unknotting no. & problem

Notation & operations	Alexander–Briggs notation Conway notation Dowker notation Flype Mutation Reidemeister move Skein relation Tabulation

Other	Alexander's theorem Berge Braid theory Conway sphere Complement Double torus Fibered Knot List of knots and links Ribbon Slice Sum Tait conjectures Twist Wild Writhe

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Slice knot

References

See also