Contents - Columbia Math

... To see why neither homotopy nor isotopy suffice, consider a knot K ⊂ R3 , and let f : S 1 → R3 denote the corresponding embedding of S 1 . I claim that there is a homotopy from f to the trivial knot g : S 1 → R3 . Indeed, just take any continuous deformation of the knot to the planar circle in R3 . ...

... To see why neither homotopy nor isotopy suffice, consider a knot K ⊂ R3 , and let f : S 1 → R3 denote the corresponding embedding of S 1 . I claim that there is a homotopy from f to the trivial knot g : S 1 → R3 . Indeed, just take any continuous deformation of the knot to the planar circle in R3 . ...

CONFIGURATION SPACE INTEGRALS AND TAYLOR TOWERS

... knots. Let Km be the space of long knots in Rm or S m , m ≥ 3. To simplify notation, we will often set K = K3 when we wish to distinguish the case of classical knots from all others. At the heart of our results are Bott-Taubes configuration space integrals [6] which are used for producing cohomology ...

... knots. Let Km be the space of long knots in Rm or S m , m ≥ 3. To simplify notation, we will often set K = K3 when we wish to distinguish the case of classical knots from all others. At the heart of our results are Bott-Taubes configuration space integrals [6] which are used for producing cohomology ...

Pizzas, Bagels, Pretzels, and Euler`s Magical χ

... What is topology? Given a set X , a topology on X is a collection T of subsets of X, satisfying the following axioms: 1. The empty set and X are in T. 2. T is closed under arbitrary union. 3. T is closed under finite intersection. ...

... What is topology? Given a set X , a topology on X is a collection T of subsets of X, satisfying the following axioms: 1. The empty set and X are in T. 2. T is closed under arbitrary union. 3. T is closed under finite intersection. ...

Fundamental Groups and Knots

... where (i,j) in Σ3 denotes the element of the permutation group which switches i and j. We can check that φ preserves the relations: φ(g3 g1 g3-1) = (3,1)(1,2)(1,3) = (2,3), φ(g1 g2 g1-1) = (1,2)(2,3)(2,1) = (3,1). Moreover, one can easily check that φ is an isomorphism. So, we can conclude that G is ...

... where (i,j) in Σ3 denotes the element of the permutation group which switches i and j. We can check that φ preserves the relations: φ(g3 g1 g3-1) = (3,1)(1,2)(1,3) = (2,3), φ(g1 g2 g1-1) = (1,2)(2,3)(2,1) = (3,1). Moreover, one can easily check that φ is an isomorphism. So, we can conclude that G is ...

# Knot theory

In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3 (in topology, a circle isn't bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.Knots can be described in various ways. Given a method of description, however, there may be more than one description that represents the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram. Any given knot can be drawn in many different ways using a knot diagram. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot.A complete algorithmic solution to this problem exists, which has unknown complexity. In practice, knots are often distinguished by using a knot invariant, a ""quantity"" which is the same when computed from different descriptions of a knot. Important invariants include knot polynomials, knot groups, and hyperbolic invariants.The original motivation for the founders of knot theory was to create a table of knots and links, which are knots of several components entangled with each other. Over six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century.To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other three-dimensional spaces and objects other than circles can be used; see knot (mathematics). Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space.