عقدة شريحة

أمثلة على العقد الشريحة

6₁

8₂₀

9₄₁

10₇₅

10₁₂₃

Ribbon knot

العقدة الشريحة slice knot، هي نوع من العقد الرياضية.

 تعريفات

In knot theory, a "knot" means an embedded circle in the 3-sphere

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

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

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

A knot ${\displaystyle{\displaystyle K\subset S^{3}}}$ is slice if it bounds a nicely embedded 2-dimensional disk D in the 4-ball.

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.

 أمثلة

The following is a list of all slice knots with 10 or fewer crossings; it was compiled using the Knot Atlas: 6₁, ${\displaystyle{\displaystyle 8_{8}}}$, ${\displaystyle{\displaystyle 8_{9}}}$, ${\displaystyle{\displaystyle 8_{20}}}$, ${\displaystyle{\displaystyle 9_{27}}}$, ${\displaystyle{\displaystyle 9_{41}}}$, ${\displaystyle{\displaystyle 9_{46}}}$, ${\displaystyle{\displaystyle 10_{3}}}$, ${\displaystyle{\displaystyle 10_{22}}}$, ${\displaystyle{\displaystyle 10_{35}}}$, ${\displaystyle{\displaystyle 10_{42}}}$, ${\displaystyle{\displaystyle 10_{48}}}$, ${\displaystyle{\displaystyle 10_{75}}}$, ${\displaystyle{\displaystyle 10_{87}}}$, ${\displaystyle{\displaystyle 10_{99}}}$, ${\displaystyle{\displaystyle 10_{123}}}$, ${\displaystyle{\displaystyle 10_{129}}}$, ${\displaystyle{\displaystyle 10_{137}}}$, ${\displaystyle{\displaystyle 10_{140}}}$, ${\displaystyle{\displaystyle 10_{153}}}$ and ${\displaystyle{\displaystyle 10_{155}}}$.

 الخصائص

Every ribbon knot is smoothly slice. An old question of Fox asks whether every smoothly slice knot is actually a ribbon knot.

The signature of a slice knot is zero.

The Alexander polynomial of a slice knot factors as a product ${\displaystyle{\displaystyle f(t)f(t^{-1})}}$ where ${\displaystyle{\displaystyle f(t)}}$ is some integral Laurent polynomial. This is known as the Fox–Milnor condition.

 انظر أيضاً

• Slice genus
• رابط شريحة