Preface ......................................................... v
1 Notation of Knots and Links .................................. 1
1.1 Basic graph theory ...................................... 1
1.2 Shadows of KLs ......................................... 10
1.2.1 Gauss and Dowker code ........................... 16
1.3 KL diagrams ............................................ 25
1.4 Reidemeister moves ..................................... 40
1.5 Conway notation ........................................ 50
1.6 Classification of KLs .................................. 59
1.7 LinKnot functions and KL notation ...................... 66
1.8 Rational world and KL invariants ....................... 69
1.8.1 Chirality of rational KLs ....................... 77
1.9 Unlinking number and unlinking gap ..................... 81
1.10 Prime and composite KLs ............................... 119
1.11 Non-invertible KLs .................................... 125
1.11.1 Tangle types ................................... 131
1.11.2 Non-invertible pretzel knots ................... 136
1.11.3 Non-invertible arborescent knots ............... 140
1.11.4 Non-invertible polyhedral knots ................ 142
1.12 Reduction of Я-tangles ................................ 145
1.12.1 KLs with unlinking number one .................. 148
1.13 Braids ................................................ 157
1.13.1 KLs and braids ................................. 161
1.14 Braid family representatives .......................... 165
1.14.1 Applications of minimum braids and braid
family representatives ......................... 179
1.15 More KL invariants .................................... 182
1.16 Borromean links ....................................... 187
2 Recognition and Generation of Knots and Links ............... 195
2.1 Recognition of KLs .................................... 195
2.1.1 Group of KL ..................................... 201
2.2 Polynomial invariants ................................. 207
2.3 Vassiliev invariants .................................. 219
2.4 Experimenting with KLs ................................ 225
2.5 Derivation and classification of KLs .................. 227
2.6 Basic polyhedra and polyhedral KLs .................... 241
2.7 Basic polyhedra and non-algebraic tangles ............. 268
2.7.1 Generalized tangles ............................ 282
2.7.2 n-tangles and basic polyhedra .................. 283
2.7.3 Non-algebraic tangle compositions and
component algebra .............................. 295
2.8 KL tables ............................................. 303
2.8.1 Non-alternating and almost alternating KLs ..... 307
2.9 Projections of KLs and chirality ...................... 311
2.10 Families of undetectable KLs .......................... 341
2.10.1 Detecting chirality of KLs by polynomial
invariants ..................................... 356
2.11 A dream— new KL tables ................................ 363
3 History of Knot Theory and Applications of Knots and
Links ...................................................... 375
3.1 History of knot theory ................................ 375
3.2 Mirror curves ......................................... 383
3.2.1 Tamil treshold designs ......................... 384
3.2.2 Tchokwe sand drawings .......................... 385
3.2.3 Construction of mirror curves .................. 388
3.2.4 Enumeration of mirror curves ................... 393
3.2.5 Lunda designs .................................. 395
3.2.6 Polyominoes .................................... 395
3.2.7 KLs and mirror curves .......................... 399
3.2.8 Mirror curves on different surfaces ............ 400
3.2.9 Mirror curves in art ........................... 401
3.2.10 KLs and self-avoiding curves ................... 416
3.3 KLs and fullerenes .................................... 426
3.3.1 General fullerenes, graphs, symmetry and
isomers ........................................ 427
3.3.2 5/6 fullerenes ................................. 428
3.3.3 Knot theory and fullerenes ..................... 430
3.3.4 Nanotubes, conical and biconical fullerenes
and their symmetry ............................. 436
3.3.5 Fullerenes on other surfaces ................... 441
3.4 KLs and logic ......................................... 443
3.5 Waveforms ............................................. 449
3.6 Knot automata ......................................... 453
Bibliography .................................................. 459
Index ......................................................... 475
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