Abstract
In this section we introduce spinning construction, which is a method of constructing 2-knots from knots.
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Notes
- 1.
Along a straight arc in \({\mathbb R}^2 = \{ (x, y, 0) \}\) or along a topographic meridian in \(S^2\).
- 2.
Refer to D.L. Goldsmith [44] and R.A. Litherland [111].
- 3.
Refer to E.C. Zeeman [184].
- 4.
In p. 314 of R.A. Litherland [111], the deformation group is defined as modulo pseudo-isotopy. Our definition is different from [111]. For classical dimension, i.e., k is a tangle, these two definitions coincide.
- 5.
This is based on a well-known fact called Alexander’s trick. Refer to R.A. Litherland [111].
- 6.
In p. 83 of [76], it is stated that Fox’s rolling is a deform-spinning with \(\rho \). This is incorrect. Fox’s rolling is a symmetry-spinning.
- 7.
The element \(\sigma _g\) depends on \(\overrightarrow{g}\). However, \(\sigma _g\) is determined from g modulo the peripheral subgroup of \(\mathcal{D}(B^3, k)\), that is generated by \(\tau \) and \(\rho \).
- 8.
A 2-knot F in \(S^4\) is called fibered if the exterior E(F) admits a structure of a fiber-bundle over \(S^1\).
- 9.
This theorem is due to T.M. Price and D.M. Roseman [138].
- 10.
- 11.
Refer to T.M. Price [137].
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Kamada, S. (2017). Spinning Construction. In: Surface-Knots in 4-Space. Springer Monographs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-10-4091-7_6
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DOI: https://doi.org/10.1007/978-981-10-4091-7_6
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