Visualization Of The Knot

Knot theory is a branch of mathematics, which studies the properties in the continuous transformation of one or more strings. When there's just one loop, it's called a knot. When there's more than one loop, it's called a link and the individual loops are called components of the link. A picture of a knot is called a knot diagram or knot projection. A place where parts of the loop cross over is called a crossing. The simplest knot is the unknot or trivial knot, which can be represented by a loop with no crossings.The big problem in knot theory is finding out whether two knots or links are the same or different. Two knots or links are regarded as being the same if they can be moved about in space, without cutting, to look exactly like each other. Such a movement is called an ambient isotopy - the ambient refers to moving the knot through 3-dimensional space, and isotopy is a scary word from topology for the continuous deformation of an object without cutting it or letting it pass through itself.In this paper, we will study the surfaces of knot spanning. There are two types of knot spanning surface which is bounded by knot: orientable surface and non-orientable one. The orientable surface is named Seifert surface and the genus of knots is defined the minimal genus of the Seifert surface. The introduction of these surfaces is shown in every text book on knot theory, but from these it is hard to understand their shape and structure. In this paper, the visualization of such surfaces is discussed.The visualization of knot spanning surfaces is very helpful to studying the genus and calculating isotopy invariants. It is also good for chemical industry and can be used for educational purpose.