Abstract
In this chapter we discuss knot concordance and cobordism. Throughout this chapter we assume that knots and links are oriented, and surfaces embedded in 4-manifolds are smooth or PL and locally flat.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This is the reason to call it a slice knot.
- 2.
Refer to Problem No. 25 of R.H. Fox [38].
- 3.
Recall the notation from Chap. 2. For an oriented knot K, we denote by \(-K\) the same knot K with the reversed orientation, and by K! the mirror image of K.
- 4.
Concordance is also called a knot cobordism.
- 5.
Refer to C. McA. Gordon [47].
- 6.
For a slice knot K, the Alexander polynomial \(\Delta (t)\) can be expressed as \(\Delta (t) = \pm t^n f(t) f(t^{-1})\) for some polynomial f(t) and an integer n (cf. Corollary 12.2.13 of A. Kawauchi [94]). The Alexander polynomial of a figure-eight is \(\Delta (t) = t^2 - t +1\), which cannot be expressed in such a form.
- 7.
It is also called a cobordism or a knot (or link) cobordism, cf. A. Kawauchi [94].
- 8.
Hint: (1) Note that K can be transformed to \(K'\) by a finite number of unknotting operations and ambient isotopies of \({\mathbb R}^3\). (2) Use a cobordism connecting K and a trivial knot.
- 9.
A survey of knot theory which focuses on the development from Goeritz matrices to quasi-alternating links by J.H. Przytycki [140] might be helpful for study.
- 10.
Let \(\beta _1(X, A)\) denote the first Betti number of a manifold pair (X, A), the rank of the free part of \(H_1(X, A).\)
- 11.
When we put K in the boundary of \(B^4\), the 4-ball genus is the minimum genus among compact connected oriented surfaces in \(B^4\) with boundary K.
- 12.
Refer to M. Khovanov [101] and E.S. Lee [106].
- 13.
This is proved in O. Plamenevskaya [135] and A. Shumakovitch [166]. It is still valid when we replace s(K) with \(2 \tau (K)\) in the right-hand side (refer to C. Livingston [113]).
- 14.
These are Theorems 1.1 and 1.3 of T. Kawamura [92] (cf. [91]), which are also valid when we replace s(K) with \(2 \tau (K)\). Refer to Corollary 1.11 of A. Lobb [115] (cf. [114]).
- 15.
Refer to L. Rudolph [155] and T. Nakamura [128]. In particular, the equality holds for a positive diagram D.
- 16.
Refer to T. Kawamura [90].
- 17.
Refer to T. Kawamura [92] (Theorems 1.1, 1.5 and 1.4).
- 18.
Refer to T. Kawamura [92] (Corollary 1.2).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Kamada, S. (2017). Knot Concordance. In: Surface-Knots in 4-Space. Springer Monographs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-10-4091-7_7
Download citation
DOI: https://doi.org/10.1007/978-981-10-4091-7_7
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-4090-0
Online ISBN: 978-981-10-4091-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)