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dubious
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i cannot find a proof anywhere to show that the linking number is always an integer! can someone please point me in the right direction (or give a proof of their own)! thanks.
Knot theory is a branch of mathematics that studies the properties of mathematical knots, which are closed loops in three-dimensional space. The linking number is a numerical invariant that measures the degree of entanglement of two knots.
The linking number is calculated by counting the number of times one knot passes through or "links" with the other knot. It can be visualized by imagining one knot as a rubber band and the other as a piece of string, and counting the number of times the rubber band passes over or under the string.
The linking number is an important topological invariant in knot theory, meaning that it remains the same even if the knots are deformed or stretched. It allows for the classification and distinction of different types of knots and helps to understand their properties and relationships.
Yes, the linking number can be negative. This occurs when one knot passes through the other in the opposite direction than it is counted, resulting in a negative number. The absolute value of the linking number remains the same, regardless of its sign.
The linking number has applications in various fields, including biology, chemistry, and physics. For example, it is used in the study of DNA and protein structures, as well as in the analysis of fluid dynamics and superconductivity. It also has practical applications in knot tying and the design of secure knots for various purposes.