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

By Seiichi Kamada

This introductory quantity offers the fundamentals of surface-knots and similar issues, not just for researchers in those parts but in addition for graduate scholars and researchers who're now not accustomed to the field.
Knot idea is without doubt one of the such a lot lively examine fields in glossy 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 larger dimensions. Surface-knots or surface-links are closed surfaces (two-dimensional manifolds) in Euclidean 4-space, that are concerning two-dimensional braids and 4-manifolds. Surface-knot concept treats not just closed surfaces but in addition surfaces with limitations in 4-manifolds. for instance, knot concordance and knot cobordism, that are additionally vital gadgets in knot thought, are surfaces within the product house of the 3-sphere and the interval.
Included during this publication are fundamentals of surface-knots and the similar subject matters of classical knots, the movie approach, floor diagrams, deal with surgical procedures, ribbon surface-knots, spinning building, knot concordance and 4-genus, quandles and their homology idea, and two-dimensional braids.

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Let D be a maximal disk of F in the level at t = d . Consider an arc α in R4 connecting an interior point q of D and a point q in R3 × {d} such that α ∩ F = {q} and α ∩ R3 × {t} is a point for each t ∈ [d , d]. ) Let F1 be a surface in R3 × [a, ∞) obtained this way. Then F ∩ R3 × [a, c] = F1 ∩ R3 × [a, c], and F1 has no critical points or critical disks in R3 × (c, d), and all maximal dasks are in R3 × {d}. 5, F1 is ambient isotopic to a surface F2 such that F2 ∩ R3 × [a, c] = F ∩ R3 × [a, c], F2 ∩ R 3 × (c, d) = L × (c, d) for a link L, and all maximal disks are in R3 × {d}.

Rolfsen [150]. A simple loop C on T is said to be of type-(p,q) if [C] = p[m] + q[l] = 0 ∈ H1 (T ). A knot is called a torus knot of type-( p, q) if it is equivalent to a knot on T that is of type-( p, q) (Fig. 33). A link L with μ components is called a torus link of type-(a, b) if a/μ and b/μ are co-prime integers and L is equivalent to a link on T each of whose components is of type-(a/μ, b/μ). A torus knot or a torus link of type-(a, b) is denoted by T (a, b) in this book. length n and the sequence (a1 , a2 , .

The HOMFLY-PT polynomial28 PL ( , m) is a link invariant that takes values in Z[ , −1 , m, m −1 ] and satisfies PO ( , m) = 1, PL + ( , m) + −1 PL − ( , m) + m PL 0 ( , m) = 0. H. Conway [29], where it was called the potential function and the relathion between the Alexander polynomial and the potential funtion was given there. R. Jones [60, 61]. H. Kauffman [87, 88] introduced a state model for the Jones polynomial. 28 HOMFLY is the initials of the authors, P. Freyd, D. Yetter, J. R. Lickorish, K.

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