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# 数学代写|扭结理论代写Knot Theory代考|MATH332 The problem of finding geometric diagrams

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## 数学代写|扭结理论代写Knot Theory代考|The problem of finding geometric diagrams

0.2.5. The problem of finding geometric diagrams. Twist knots and 2-bridge knots have standard diagrams that encode a great deal of information about the geometry of the knot. Does every knot have such a diagram? (Probably not.) Does every knot have a diagram from which we may read some geometric information?

Alternating knots are another family of knots that seem to be amenable to hyperbolic geometric techniques.

Definition 0.14. An alternating diagram is a diagram of a knot or link that has an orientation such that, when following the knot in the direction of the orientation, the crossings alternate between over and under. An alternating knot or link is a knot or link that has an alternating diagram.

We will see in the exercises in Chapter 1 that alternating knot complements decompose into pieces with the same combinatorics of the diagram. In chapters 11,12 , and 13 we will use this decomposition to determine some geometric information on the knot complement.

How useful is this work broadly? All knots with at most seven crossings have alternating diagrams. Tait began his work [Tai98] by assuming diagrams were alternating (although he did publish diagrams of eight- and ten-crossing nonalternating examples in 1877). However, the proportion of alternating knots in diagrams enumerated by crossing number rapidly drops to zero [ST98, Thi98]. As for knots enumerated by geometric triangulations, nonalternating examples seem to be even more common; a nonalternating example appears as the second knot on the list in Figure 0.14. Thus, unfortunately, alternating knots and links are not very common.

## 数学代写|扭结理论代写Knot Theory代考|The problem of determining geometric invariants

The problem of determining geometric invariants. One way of distinguishing knots is to compute invariants for each of them. If the invariants disagree, then the knots cannot be equivalent.

Several knot invariants arise classically, such as the crossing number that we encountered above. Many additional knot invariants arise through geometry. One aim of this book is to discuss such invariants, and give tools to calculate them.

One of the most straightforward knot invariants that arises in geometry is the volume of a knot. We will show in Chapter 5 that any knot complement that admits a hyperbolic structure has finite volume. Thus volumes of knots give knot invariants.

For those knots whose diagrams are particularly amenable to geometric techniques, such as twist knots, 2-bridge knots, and alternating knots, there are known methods to estimate volume using the combinatorics of the diagram. This is discussed along the way, but especially in Chapter 13 , where we bring to bear several tools in geometry to give two-sided bounds on volumes.

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