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# 数学代写|扭结理论代写Knot Theory代考|MATH5801 Problems in knot theory

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## 数学代写|扭结理论代写Knot Theory代考|Problems in knot theory

There are many open problems in knot theory, and as new mathematical fields are brought to bear upon these problems, new questions and problems arise. This section gives a few highlights of the most classical problems, and also problems that seem most amenable to geometric techniques. Probably the most long-standing problem, and also one of the most broad, is the following.

The classification problem. When do two different descriptions of knots yield equivalent knots? When do they have homeomorphic complements?

When the description of a knot is given by a diagram, this is the problem that Tait encountered while trying to list all knots with a fixed number of crossings. See Figure 0.2, which is modified from the 1884 paper [Tai 4].
There are a few moves that can be performed on a diagram that do not change the equivalence class of the underlying knot. For example, if the diagram contains a single crossing that forms a loop, as shown on the left of Figure $0.3$, that loop can be untwisted to simplify the diagram.

## 数学代写|扭结理论代写Knot Theory代考|The problem of determining geometry of the complement

Briefly, the complement of a knot is hyperbolic if and only if it admits a complete metric with all sectional curvatures equal to $-1$. We will give other equivalent definitions of hyperbolic knots in later chapters, which will often be more useful for calculations, computations, and examples.

For now, it is known that when a knot complement is hyperbolic, its hyperbolic metric is unique. That is, hyperbolic knot complements that are homeomorphic must also be isometric under any hyperbolic metrics placed upon their complements. Moreover, a large number of knots are hyperbolic, and many that are not hyperbolic decompose into hyperbolic pieces.
More precisely, consider the following families of knots.
Definition 0.8. A torus knot is a knot that can be embedded on the surface of an unknotted torus in $S^3$ (without crossings on the surface of the torus). See Figure 0.7.

By an unknotted torus, we mean the boundary of a regular neighborhood of an unknot in $S^3$, with no crossings.

Tai4

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