Preface ....................................................... vii
List of figures ................................................ xi
1 Basic knots, links and their equivalences .................... 1
1.1 Definitions and equivalences ............................ 1
1.2 Polygonal (PL), smooth (C∞)-links and knots in  3 ....... 3
1.3 Continuous links in 3 .................................. 7
1.4 Reidemeister moves and equivalences ..................... 7
1.5 Crossing number and knot tabulation .................... 10
1.6 Brief history on the knot tabulation ................... 14
2 Braids and links ............................................ 21
2.1 Braid definition ....................................... 21
2.2 Artin's classical braid group Σ = 2 ................... 25
2.3 Artin's theorem on presentation of the full braid
group .................................................. 27
2.4 Normal form of braid elements .......................... 32
2.5 Braid action on a free group ........................... 32
2.6 Alexander theorem ...................................... 36
2.7 Characterization of braid representation ............... 40
2.8 Link group via the braid presentation .................. 43
2.9 Braid groups are linear ................................ 45
2.10 Singular braids ........................................ 51
2.11 (Co)Homology of braid groups and loop spaces ........... 53
3 Knot and link invariants .................................... 57
3.1 Basic background ....................................... 57
3.2 Unknotting and unknotting number ....................... 58
3.3 Bridge number and total curvature ...................... 65
3.4 Linking number and crossing number ..................... 71
3.5 Knot group and Wirtinger presentation .................. 77
3.6 Free differential calculus ............................. 84
3.7 Magnus representations ................................. 91
3.8 Alexander ideals and Alexander polynomials ............. 98
3.9 Twisted Alexander polynomials ......................... 103
4 Jones polynomials .......................................... 111
4.1 Hecke algebra ......................................... 111
4.2 Ocneanu trace on Hn(q) ................................ 117
4.3 Two-variable Jones polynomial ......................... 119
4.4 Skein relation ........................................ 125
4.5 Uniqueness and universal property of skein invariant .. 127
4.6 Properties of Jones polynomial ........................ 131
4.7 Kauffman bracket and polynomial ....................... 140
4.8 The Tait conjectures .................................. 144
4.9 The plat approach to Jones polynomials ................ 154
4.10 Homological definition of the Jones polynomial ........ 160
5 Casson type invariants ..................................... 167
5.1 The SU(2) group ....................................... 167
5.2 The SU(2) representations ............................. 172
5.3 The SU(2) Casson invariant of integral homology
3-spheres ............................................. 180
5.4 The SU(2) Casson invariant of knots ................... 182
5.5 Representation space of knot complements .............. 187
5.6 The SU(2) Casson-Lin invariant of knots ............... 193
5.7 Surgery relation for the SU(2) Casson-Lin invariants .. 197
5.8 Other extensions of the SU(2) Casson-Lin invariants ... 208
5.9 Calculation of the SU(2) Casson-Lin invariants ........ 218
Bibliography .................................................. 223
Index ......................................................... 229
|
