Abstract
A knot is a submanifold of \({\mathbb R}^3\) that is homeomorphic to a circle. A link with \(\mu \) components means a union \(L=K_1\cup \cdots \cup K_\mu \) of mutually disjoint \(\mu \) knots \(K_1, \dots , K_\mu \). Each knot \(K_i\) is called a component of the link.
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Notes
- 1.
Refer to R.H. Crowell and R.H. Fox [30], Appendix I.
- 2.
The idea that a smooth-looking knot is PL with a bunch of edges is due to J.W. Alexander in the 1928 paper [3].
- 3.
A \(\varDelta \)-move is also called an elementary deformation in J.W. Alexander and G.B. Briggs [4]. This combinatorial move is due to K. Reidemeister [145]. Refer also to K. Reidemeister [146–148].
- 4.
This theorem is also valid for links. For a proof, refer to G. Burde and H. Zieschang [14] or A. Kawauchi [94].
- 5.
Refer to A. Kawauchi [93] and H.F. Trotter [176].
- 6.
Hint. Let \(r': {\mathbb R}^3 \rightarrow {\mathbb R}^3\) be an orientation-reversing homeomorphism. Then \(r\circ (r')^{-1}: {\mathbb R}^3 \rightarrow {\mathbb R}^3\) is an orientation-preserving homeomorphism sending \(r'(K)\) to r(K).
- 7.
The Jones polynomials (Sect. 2.7) of the trefoil knot and its mirror image are different.
- 8.
Refer to A. Kawauchi [93] and H.F. Trotter [176].
- 9.
A point p of f(K) is a multiple point if \(f^{-1}(p)\) consists of two points or more.
- 10.
The move indicated with III\(^\#\) in Fig. 2.8 is also a Reidemeister move of type III. For convenience, here we distinguish between the moves of type III and III\(^\#\).
- 11.
This is due to K. Reidemeister in the 1926 article [145]. It is also found in the 1927 article by J.W. Alexander and G.B. Briggs [4]. Refer also to Reidemeister’s book [146–148].
- 12.
For details on these generating sets, refer to M. Polyak [136].
- 13.
When \(\mu \ge 2\), if an oriented link L with \(\mu \) components has a Seifert surface such that the number of connected components of S is \(\mu \), then L is called a boundary link.
- 14.
This method was introduced by H. Seifert [165]. Note that not every Seifert surface of a knot is obtained by this method.
- 15.
The genus of a compact, connected, oriented surface means the genus of a closed surface obtained by attaching disks along the boundary components.
- 16.
When we consider surfaces S and \(S'\) in \(S^3\), infinity passing moves are not needed. Handle equivalence is also called tube equivalence in D. Bar-Natan, J. Fulman, and L.H. Kauffman [10].
- 17.
This theorem is used in J. Levine [109] to prove the uniqueness of the S-equivalence classes of Seifert surfaces (Sect. 2.6). For a proof, refer to D. Bar-Natan, J. Fulman, and L.H. Kauffman [10] or Proposition 7.2.2 of A. Kawauchi [96].
- 18.
When we consider non-orientable surface S, we do not assume that the 1-handle h is coherent with respect to an orientation of S in the definition of a handle addition. This theorem can be proved by an argument similar to the proof of Proposition 7.2.2 of A. Kawauchi [96].
- 19.
This theorem is due to H. Schubert [163]. For a link case, refer to Y. Hashizume [51].
- 20.
This is due to C.D. Papakriakopoulos [134].
- 21.
This is due to J. Levine [109]. Refer to A. Kawauchi [96] for details.
- 22.
Refer to H.F. Trotter [175], K. Murasugi [127], J. Levine [109], C. McA. Gordon and R.A. Litherland [48] for signatures of knots and links.
- 23.
When K is a knot, \(n(K)=0\).
- 24.
A Laurent polynomial is a polynomial that may have negative powers of the variable. When the coefficients are integers, we call it an integral Laurent polynomial. For example, \(3t^2 - 6 + 2t^{-1} - t^{-3}\).
- 25.
Refer to J.W. Alexander [3].
- 26.
The invariant \(\nabla _L(z)\) was defined by J.H. Conway [29], where it was called the potential function and the relathion between the Alexander polynomial and the potential funtion was given there.
- 27.
Refer to V.F.R. Jones [60, 61]. L.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. Hoste, W.B.R. Lickorish, K. Millett, and A. Ocneanu of [36] (cf. [110]) and PT is that of the authors, J.H. Przytycki and P. Traczyk of [142].
- 29.
The length n and the sequence \((a_1, a_2, \dots , a_n)\) are not determined uniquely. One may take such a sequence \((a_1, a_2, \dots , a_n)\) in even numbers. Especially, when we consider oriented 2-bridge knots and links, a sequence \((a_1, a_2, \dots , a_n)\) in even numbers is preferred.
- 30.
Refer to A. Kawauchi [94].
- 31.
Refer to A. Kawauchi [96].
- 32.
It follows from Thurston’s hyperbolization theorem. G. Perelman proved Thurston’s geometrization conjecture, that implies Thurston’s hyperbolization theorem and the Poincaré conjecture. Refer to J. Morgan and G. Tian [125, 126].
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Kamada, S. (2017). Knots. In: Surface-Knots in 4-Space. Springer Monographs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-10-4091-7_2
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