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In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R³ (also known as E³), considered up to continuous deformations (isotopies). A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term knot is also applied to embeddings of ${\displaystyle S^{j}}$ in ${\displaystyle S^{n}}$, especially in the case ${\displaystyle j=n-2}$. The branch of mathematics that studies knots is known as knot theory, and has many simple relations to graph theory.

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