摘要
扭結是跟單位圓同坯的三維空間的一個子空間。在紐結理論，如何辨別兩個扭結的異同是很重要的問題。我們將會介紹一些與扭結有關的拓樸不變量，其中包含扭結群、由赫爾伯特･塞弗特建構的賽佛特表面、詹姆斯･韋德爾･亞歷山大在1923年所做的亞歷山大多項式、環繞數及沃恩•瓊斯在1984年發現的康威多項式以及瓊斯多項式。其中，康威多項式又稱為康威─亞歷山大多項式，在亞歷山大的論文中曾證明其滿足糾結關係，而瓊斯只是發現了類似的關係式而已。我們會使用以上提及的不變量來處理扭結的等價關係。 給定一個扭結，我們將其扭結群定義為三維空間中扭結補集的基本群，這是我們第三章會提及的概念。此外，我們也會說明如何去建構扭結群，並計算一些相關例子。雖然扭結群很容易計算，但這個不變量卻無法去分辨一些不同的扭結，舉例來說，祖母結以及平結，因此我們需要其他的扭結不變量來協助我們去分辨不同的扭結。在第四章及第五章，我們會先建構扭結的賽佛特表面，並且去計算其虧格以及其亞歷山大多項式，後者在適當投影下，其投影的選擇並不影響亞歷山大多項式，因此亦為扭結不變量。第六章我們將鍵結列為重點討論。我們會介紹萊德邁斯特移動、環繞數及第一同調群，並用於推導出康威多項式、亞歷山大多項式以及雙覆蓋鍵環多項式。康威多項式和亞歷山大多項式實際上可以互相轉換，因此他們能辨別的扭結狀況是一樣的。他們亦可視為雙覆蓋鍵環多項式的特例。雖然雙覆蓋鍵環多項式是最一般的扭結不變量，有些扭結及其變種卻不能被他辨別出來。 最後一章我們將使用一個軟體:塞弗特視圖，我們將會用這個軟體來畫一些論文中提及的扭結及其賽佛斯表面。
並列摘要
A knot is a subspace of the three-dimensional Euclidean space which is homeomorphic to the unit circle. How to determine whether two knots are equivalent or not is a fundamental question in knot theory. We will introduce some topological invariants for knots, including knot groups, Seifert surfaces named after German mathematician Herbert Seifert, Alexander polynomials discovered by James Waddell Alexander II in 1923, linking numbers, Conway polynomials and Jones polynomials discovered by Vaughan Jones in 1984. The Conway polynomials, also called Alexander-Conway polynomials, was proved to satisfy skein relations by Alexander, and John Conway rediscovered it in a different form. We will use those invariants to investigate the equivalence problem of knots. The knot group of a given knot which is the fundamental group of the complement of the given knot, will be introduced in the third section. We will construct it in detail and calculate some examples. Although they are easy to calculate, they cannot distinguish some different knots, for example, the granny and the square knots, so we need other knot invariants to help us to distinguish different knots. In the fourth and fifth section, we first construct Seifert surfaces for knots, calculate their genus and then construct Alexander polynomials, another knot invariant which is independent of the choice of nice projections of knots. In the sixth section, we focus on invariants of links. We introduce Reidemeister moves, linking numbers and first homology groups and use them to study Conway polynomials, Alexander polynomials and HOMFLY polynomials. After some transformation, Conway polynomials can be transfered to Alexander polynomials, so the knots which they can distinguish are the same. They can both be regarded as special cases of HOMFLY polynomials, which is the most general knot invariant among them, but still, HOMFLY polynomials are unable to distinguish some knots from their mutations. In the last section, we use a software called Seifertview, which makes knots and their Seifert surfaces visible, and we will present some knots which we mention in this thesis.