We show that for any link L, there exists a Seifert surface for L that is obtained by successively plumbing flat annuli to a disk D, where the gluing regions are all in D. This furnishes a new way of coding links. We also present an algorithm to read the code directly from a braid presentation.

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We present a practical algorithm to determine the minimal genus of non-orientable spanning surfaces for 2-bridge knots, called the crosscap numbers. We will exhibit a table of crosscap numbers of 2-bridge knots up to 12 crossings (all 362 of them).

In this paper, we define the Gordian complex of knots, which is a simplicial complex whose vertices consist of all oriented knot types in the 3-sphere. We show that for any knot K, there exists an infinite family of distinct knots containing K such that any pair (Ki, Kj) of the member of the family, the Gordian distance dG(Ki, Kj) = 1.

Necessary and sufficient conditions are given for a satellite knot to be fibered. Any knotk embeds in an unknotted solid torusṼ with arbitrary winding number in such a way that no satellite knot with pattern (Ṽ ,k) is fibered. In particular, there exist nonfibered satellite knots with fibered pattern and companion knots and nonzero winding number.