Download e-book for iPad: Surface-Knots in 4-Space: An Introduction by Seiichi Kamada

By Seiichi Kamada

ISBN-10: 9811040915

ISBN-13: 9789811040917

This introductory quantity presents the fundamentals of surface-knots and comparable issues, not just for researchers in those components but additionally for graduate scholars and researchers who're now not conversant in the field.
Knot conception is without doubt one of the so much lively examine fields in smooth arithmetic. Knots and hyperlinks are closed curves (one-dimensional manifolds) in Euclidean 3-space, and they're concerning braids and 3-manifolds. those notions are generalized into greater dimensions. Surface-knots or surface-links are closed surfaces (two-dimensional manifolds) in Euclidean 4-space, that are regarding two-dimensional braids and 4-manifolds. Surface-knot conception treats not just closed surfaces but additionally surfaces with barriers in 4-manifolds. for instance, knot concordance and knot cobordism, that are additionally vital gadgets in knot conception, are surfaces within the product house of the 3-sphere and the interval.
Included during this publication are fundamentals of surface-knots and the comparable issues of classical knots, the movie procedure, floor diagrams, deal with surgical procedures, ribbon surface-knots, spinning building, knot concordance and 4-genus, quandles and their homology concept, and two-dimensional braids.

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Example text

27 The relation at a crossing: x −1 j xi x j = xk xi ai aj xj xk ak Take a base point p of G(K ) in the open upper-half space {(x, y, z) ∈ R3 | z > 0}. For each arc ai (i = 1, . . , m), let Bi be a meridian disk at an interior point of ai . Take a point qi on ∂ Bi in the open upper-half space. Let αi be a straight path from qi to p. Let m i be a positive meridian loop that starts at qi and goes along ∂ Bi . Then αi−1 m i αi is a loop in R3 \ K with base point p. Put xi := [αi−1 m i αi ] ∈ G(K ) = π1 (R3 \ K , p).

P. J. Sanderson [152]. 48 3 Motion Pictures (1) For each s ∈ [0, 1], ψs : R3 × [0, 1] → R3 × [0, 1] preserves the t-levels. (2) For each s ∈ [0, 1], ψs |R3 × {1} is the identity map. (3) ψ1 (F) = L + × [0, 1]. 7 (Horibe and Yanagawa’s lemma) Let F be a 2-link with μ components satisfying the following: ⎧ + D ∪ · · · ∪ Dμ+ ⎪ ⎪ ⎨ 1 L pr(F ∩ R3 × {t}) = ⎪ D1− ∪ · · · ∪ Dμ− ⎪ ⎩ ∅ (t = b) (a < t < b) (t = a) (t < a or b < t). Here a < b, L is a trivial link with μ components, and each of {D1− , . .

Suppose that the following two conditions are satisfied: (1) There exists a real number c with c > a such that F ∩ R3 × [a, c) = F ∩ R3 × [a, c). (2) All maximal disks of F and F are in R3 × {c}, and there are no critical points or critical bands except the maximal disks. Also F ∩ R3 × (c, ∞) = F ∩ R3 × (c, ∞) = ∅. Then there exists an ambient isotopy (h s | s ∈ [0, 1]) of R3 × [a, ∞) rel R3 × {a} carrying F to F . Moreover, for a given positive number δ with a < c − δ, we may assume (h s | s ∈ [0, 1]) to be rel R3 × [a, c − δ].

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Surface-Knots in 4-Space: An Introduction by Seiichi Kamada

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