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| | I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. Playing badminton is a thing that he is completely addicted to. Office supervising is my occupation. Some time ago she selected to live in Alaska and her parents live nearby.<br><br>my site ... best psychic; [http://chungmuroresidence.com/xe/reservation_branch2/152663 http://chungmuroresidence.com/xe/reservation_branch2/152663], |
| A '''slice knot''' is a type of [[knot theory|mathematical knot]]. It helps to remember that in [[knot theory]], a "knot" means an embedded [[circle]] in the [[3-sphere]]
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| :<math>S^3 = \{\mathbf{x}\in \mathbb{R}^4 \mid |\mathbf{x}|=1 \}</math>
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| and that the 3-sphere can be thought of as the boundary of the four-dimensional [[ball (mathematics)|ball]] | |
| :<math>B^4 = \{\mathbf{x}\in \mathbb{R}^4 \mid |\mathbf{x}|\leq 1 \}.</math>
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| A knot <math>K\subset S^3</math> is '''slice''' if it bounds a nicely embedded disk ''D'' in the 4-ball.
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| What is meant by "nicely embedded" depends on the context, and there are different terms for different kinds of slice knots. If ''D'' is [[smooth function|smoothly]] embedded in ''B<sup>4</sup>'', then ''K'' is said to be '''smoothly slice'''. If ''K'' is only [[locally flat]] (which is weaker), then ''K'' is said to be '''topologically slice'''.
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| Any [[ribbon knot]] is smoothly slice.
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| An old question of [[Ralph Fox|Fox]] asks whether every slice knot is actually a ribbon knot.
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| The [[signature of a knot|signature]] of a slice knot is zero.
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| The Alexander polynomial of a slice knot factors as a product <math>f(t)f(t^{-1})</math> where <math>f(t)</math> is some integral Laurent polynomial. This is known as the '''Fox–Milnor condition'''.
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| The following is a list of all slice knots with 10 or fewer crossings; it was compiled using the [http://katlas.math.toronto.edu/ Knot Atlas]{{full|date=April 2013}}:
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| [[6 1 knot|6<sub>1</sub>]], <math>8_8</math>, <math>8_9</math>, <math>8_{20}</math>, <math>9_{27}</math>, <math>9_{41}</math>, <math>9_{46}</math>, <math>10_3</math>, <math>10_{22}</math>, <math>10_{35}</math>, <math>10_{42}</math>, <math>10_{48}</math>, <math>10_{75}</math>, <math>10_{87}</math>, <math>10_{99}</math>, <math>10_{123}</math>, <math>10_{129}</math>, <math>10_{137}</math>, <math>10_{140}</math>, <math>10_{153}</math> and <math>10_{155}</math>.
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| ==See also==
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| *[[Slice genus]]
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| *[[Slice link]]
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| {{Knot theory|state=collapsed}}
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| [[Category:Slice knots and links| ]]
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| {{knottheory-stub}}
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