Notes for AKT-140108/0:08:12 - Revision history

Leo algknt at 05:19, 31 May 2018

2018-05-31T05:19:16Z

← Older revision		Revision as of 01:19, 31 May 2018
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	'''18S-AKT Question:''' Why is this sum divisible by 2? Why the $\frac{1}{2}$?		'''18S-AKT Question:''' Why is this sum divisible by 2? Why the $\frac{1}{2}$?

	The factor $\frac{1}{2}$ I think is as a result of the projection of the link unto the plane making each sign appear twice.		'''Attempt 1''' The factor $\frac{1}{2}$ I think is as a result of the projection of the link unto the plane making each sign appear twice.

	'''Jordan Curve Theorem.''' If $C$ is a simple closed curve in $\mathbb{R}^2$, then the complement ${\mathbb R}^2\setminus C$ has two components, the interior and the exterior, with $C$ the boundary of each.		'''Jordan Curve Theorem.''' If $C$ is a simple closed curve in $\mathbb{R}^2$, then the complement ${\mathbb R}^2\setminus C$ has two components, the interior and the exterior, with $C$ the boundary of each.


	The Jordan curve theorem implies that two distinct components in a diagram for a link $L$ intersect an even number of times. Hence we add up an even		'''Attempt 2'''The Jordan curve theorem implies that two distinct components in a diagram for a link $L$ intersect an even number of times. Hence we add up an even
	number of $\pm 1$'s in the computation of $lk(L)$, which yields an even number. It is always an integer. This is why we have a factor of $\frac12$.		number of $\pm 1$'s in the computation of $lk(L)$, which yields an even number. It is always an integer. This is why we have a factor of $\frac12$.


			'''Attempt 3''' '''Improved'''
			The Jordan curve theorem requires that we have simple closed curves and in general the components of a link are not simple closed curves. However, in the definition of the linking number, only the crossings between the components count. So we can smooth out crossings that are not between the components and this will not affect the linking number; we will then get components that are simple closed curves. From this argument, we can then apply the Jordan curve theorem to get the desired result.

Drorbn at 17:32, 23 May 2018

2018-05-23T17:32:29Z

← Older revision		Revision as of 13:32, 23 May 2018
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	The Jordan curve theorem implies that two distinct components in a diagram for a link $L$ intersect an even number of times. Hence we add up an even		The Jordan curve theorem implies that two distinct components in a diagram for a link $L$ intersect an even number of times. Hence we add up an even
	number of $\~~pm1~~$’s in the computation of $lk(L)$, which yields an even number. It is always an integer. This is why we have a factor of $\frac12$.		number of $\pm 1$'s in the computation of $lk(L)$, which yields an even number. It is always an integer. This is why we have a factor of $\frac12$.

Drorbn at 17:31, 23 May 2018

2018-05-23T17:31:02Z

← Older revision		Revision as of 13:31, 23 May 2018
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	The factor $\frac{1}{2}$ I think is as a result of the projection of the link unto the plane making each sign appear twice.		The factor $\frac{1}{2}$ I think is as a result of the projection of the link unto the plane making each sign appear twice.

	'''Jordan Curve Theorem.''' If $C$ is a simple closed curve in $\mathbb{R}^2$, then the complement $R^2-\setminus C$ has two components, the interior and the exterior, with $C$ the boundary of each.		'''Jordan Curve Theorem.''' If $C$ is a simple closed curve in $\mathbb{R}^2$, then the complement ${\mathbb R}^2\setminus C$ has two components, the interior and the exterior, with $C$ the boundary of each.

	The Jordan curve theorem implies that two distinct components in a diagram for a link $L$ intersect an even number of times. Hence we add up an even		The Jordan curve theorem implies that two distinct components in a diagram for a link $L$ intersect an even number of times. Hence we add up an even

Drorbn at 17:30, 23 May 2018

2018-05-23T17:30:01Z

← Older revision		Revision as of 13:30, 23 May 2018
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	18S-AKT Question: Why is this sum divisible by 2? Why the $\frac{1}{2}$?		'''18S-AKT Question:''' Why is this sum divisible by 2? Why the $\frac{1}{2}$?

	The factor $\frac{1}{2}$ I think is as a result of the projection of the link unto the plane making each sign appear twice.		The factor $\frac{1}{2}$ I think is as a result of the projection of the link unto the plane making each sign appear twice.

		⚫	'''Jordan Curve Theorem.''' If $C$ is a simple closed curve in $\mathbb{R}^2$, then the complement $R^2-\setminus C$ has two components, the interior and the exterior, with $C$ the boundary of each.
	\textbf{Jordan Curve Theorem}
⚫	If $C$ is a simple closed curve in \mathbb{R}^2, then the complement R^2-J has two components, the interior and the exterior, with $C$ the boundary of each.

	The Jordan curve theorem implies that two distinct components in a diagram for a link $L$ intersect an even number of times. Hence we add up an even		The Jordan curve theorem implies that two distinct components in a diagram for a link $L$ intersect an even number of times. Hence we add up an even
	number of $\pm1$’s in the computation of lk(L), which yields an even number. It is always an integer. This is why we have ~~afactor~~ of $\~~frac{1}{2}~~$.		number of $\pm1$’s in the computation of $lk(L)$, which yields an even number. It is always an integer. This is why we have a factor of $\frac12$.

Leo algknt at 17:25, 23 May 2018

2018-05-23T17:25:09Z

← Older revision		Revision as of 13:25, 23 May 2018
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	\textbf{Jordan Curve Theorem}		\textbf{Jordan Curve Theorem}
	If $C$ is a simple closed curve in \mathbb{R}^2, then the complement R^2-J has two components: the interior and the exterior, with $C$ the boundary of each.		If $C$ is a simple closed curve in \mathbb{R}^2, then the complement R^2-J has two components, the interior and the exterior, with $C$ the boundary of each.

	The Jordan curve theorem implies that two distinct components in a diagram for L intersect an even number of times. Hence we add up an even		The Jordan curve theorem implies that two distinct components in a diagram for a link $L$ intersect an even number of times. Hence we add up an even
	number of $\pm$ ~~1’s~~ in the computation of lk(L), which yields an even number		number of $\pm1$’s in the computation of lk(L), which yields an even number. It is always an integer. This is why we have afactor of $\frac{1}{2}$.
	and hence an integral linking number.

Leo algknt at 16:59, 23 May 2018

2018-05-23T16:59:06Z

← Older revision		Revision as of 12:59, 23 May 2018
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	The factor $\frac{1}{2}$ I think is as a result of the projection of the link unto the plane making each sign appear twice.		The factor $\frac{1}{2}$ I think is as a result of the projection of the link unto the plane making each sign appear twice.

			\textbf{Jordan Curve Theorem}
			If $C$ is a simple closed curve in \mathbb{R}^2, then the complement R^2-J has two components: the interior and the exterior, with $C$ the boundary of each.

			The Jordan curve theorem implies that two distinct components in a diagram for L intersect an even number of times. Hence we add up an even
			number of $\pm$ 1’s in the computation of lk(L), which yields an even number
			and hence an integral linking number.

Leo algknt at 07:31, 9 May 2018

2018-05-09T07:31:36Z

← Older revision		Revision as of 03:31, 9 May 2018
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	18S-AKT Question: Why is this sum divisible by 2? Why the $\frac{1}{2}$?		18S-AKT Question: Why is this sum divisible by 2? Why the $\frac{1}{2}$?

	The factor $\frac{1}{2}$ I think is as a result of the projection of the ~~knot~~ unto the plane making each sign appear twice.		The factor $\frac{1}{2}$ I think is as a result of the projection of the link unto the plane making each sign appear twice.

Leo algknt: 20180509: Reason for the factor half

2018-05-09T07:30:07Z

20180509: Reason for the factor half

← Older revision		Revision as of 03:30, 9 May 2018
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	18S-AKT Question: Why is this sum divisible by 2? Why the $\frac{1}{2}$?		18S-AKT Question: Why is this sum divisible by 2? Why the $\frac{1}{2}$?

			The factor $\frac{1}{2}$ I think is as a result of the projection of the knot unto the plane making each sign appear twice.

Drorbn: Created page with "18S-AKT Question: Why is this sum divisible by 2? Why the $\frac{1}{2}$?"

2018-05-03T19:01:13Z

Created page with "18S-AKT Question: Why is this sum divisible by 2? Why the $\frac{1}{2}$?"

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18S-AKT Question: Why is this sum divisible by 2? Why the $\frac{1}{2}$?