Abstract
Let D be a diagram of a knot (or a link), and let \(\mathrm{Arc}(D)= \{a_1, \dots , a_m\}\) be the set of arcs of D.
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Notes
- 1.
Refer to J.H. Przytycki [139].
- 2.
Fox’s original idea of a 3-coloring is a representation \(\pi _1({\mathbb R}^3 \setminus K) \rightarrow \mathcal{S}_3\) of the knot group to the symmetric group \(\mathcal{S}_3\) such that meridian elements are mapped to transpositions (12), (13), (23). Refer to R.H. Crowell and R.H. Fox [30] (p. 133).
- 3.
In [76], motion pictures corresponding to Roseman moves are illustrated. In order to describe an n-coloring of D, we may consider coherent n-colorings of the cross-sections.
- 4.
The notion of a kei was introduced by M. Takasaki [169] in 1943 as an algebra that is an abstraction of symmetric transformations. D. Joyce [64] introduced it as a kei in English.
- 5.
Refer to R. Fenn and C. Rourke [33] for details on racks, and refer to P. Dehornoy [31] for details on right self-distributive systems.
- 6.
The suffix \(\mathrm{L}\) (or \(\mathrm{R}\)) means the action is from the left (or the right).
- 7.
When X is a finite quandle, a subset A is closed under \(*\) if and only if it is closed under \(\overline{*}\). When X is not a finite quandle, it is not true in general that A is closed under \(\overline{*}\) if it is closed under \(*\), [79].
- 8.
We also denote by e or 1 the identity element of the free group.
- 9.
Hint: Assertion 2(m): f and g are the same on E(S, X; m). Prove this.
- 10.
For presentations of groups, refer to R.H. Crowell and R.H. Fox [30].
- 11.
In R. Fenn and C. Rourke [33], \(\langle \langle R \rangle \rangle _\mathrm{rack}\) is defined by (E1), (E2)(E3)(R1\(''\))(R2). In the definitions of (R1\(''\)) and (R2) in [33] dual operations should be considered.
- 12.
The first \(\langle \langle R \rangle \rangle _\mathrm{qdle}\) in this equality is a subset of \(FR(S) \times FR(S)\) and the second one is a subset of \(FQ(S) \times FQ(S)\).
- 13.
In this book, we use the symbol \(\langle \, S \mid R \, \rangle \) to present a rack (or a quandle) or a presentation. When we need to distinguish them, we should use different notations. For example, (SÂ :Â R) for a presentation and |SÂ :Â R| for a rack (or a quandle) determined by the presentation (SÂ :Â R).
- 14.
A map \(h: S \rightarrow S'\) induces a homomorphism \(h: FR(S) \rightarrow FR(S')\). The product \(h \times h: FR(S) \times FR(S) \rightarrow FR(S') \times FR(S')\) is also denoted by h for simplicity.
- 15.
The isomorphism \(\phi h^{-1}: \langle \, S \mid R' \, \rangle \rightarrow X\) and the isomorphism \(\phi : \langle \, S \mid R \, \rangle \rightarrow X\) are induced from the same map \(\phi h^{-1} = \phi : S \rightarrow X\). Thus, instead of \(\phi '\), we usually write \(\phi : \langle \, S \mid R' \, \rangle \rightarrow X\).
- 16.
This is proved in R. Fenn and C. Rourke [33] for the case of racks. This theorem corresponds to Tietze’s theorem on presentations of groups (cf. R.H. Crowell and R.H. Fox [30]).
- 17.
The symbol \(\mathrm{As}(X)\) is due to S. Matveev [120] and R. Fenn and C. Rourke [33]. D. Joyce [64] used the symbol \(\mathrm{Adconj} X\) (Proposition 8.8.6). The associated group is also called the adjoint group and denoted also by \(G_X\).
- 18.
This proposition, in the case of a rack, is given in Sect. 6 of D. Joyce [64]. In Proposition 2.1 of R. Fenn and C. Rourke [33], condition (1) is not written explicitly. However, as shown in Remark 8.8.5, condition (1) is essential for (3).
- 19.
When X is the knot quandle of a knot K, the associated group \(\mathrm{As}(X)\) is isomorphic to the knot group of K (Sect. 8.9), which is not a free group unless K is a trivial knot.
- 20.
This is the original idea of the associated group \(\mathrm{As}(X)\) by D. Joyce [64] and S. Matveev [120].
- 21.
\((D, \alpha )\) is homotopic to \((D', \alpha ')\) if there exists a 1-parameter family \(( (D_s, \alpha _s) \mid s \in [0, 1])\) with \((D, \alpha )= (D_0, \alpha _0)\) and \((D', \alpha ') = (D_1, \alpha _1)\) such that for each \(s \in [0, 1]\), \(D_s\) is a meridian disk of K and \(\alpha _s\) is a path in E(K) starting from a point of \(\partial D_s\) and terminating at the base point p.
- 22.
Refer to D. Joyce [64] and S. Matveev [120]. A similar result holds when K and \(K'\) are non-splittable oriented links.
- 23.
Refer to K. Tanaka [172].
- 24.
For a proof, refer to R. Fenn and C. Rourke [33], D. Joyce [64], S. Matveev [120].
- 25.
It is not necessary to consider triple points and branch points of D. This theorem is proved by an argument similar to that in [75].
- 26.
Refer to R. Fenn and C. Rourke [33].
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Kamada, S. (2017). Quandles. In: Surface-Knots in 4-Space. Springer Monographs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-10-4091-7_8
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