Complexity: knots, colourings and counting
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge u.a.
Cambridge Univ. Press
1993
|
| Edition: | 1. publ. |
| Series: | London Mathematical Society: London Mathematical Society lecture note series
186 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VIII, 163 S. graph. Darst. |
| ISBN: | 0521457408 |
Staff View
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Record in the Search Index
| _version_ | 1804122639089795072 |
|---|---|
| adam_text | Contents
1 The complexity of enumeration 1
1.1 Basics of complexity 1
1.2 Counting problems 3
1.3 # P-complete problems 6
1.4 Decision easy, counting hard 10
1.5 The Permanent 11
1.6 Hard enumeration problems not thought to be #P-complete. 14
1.7 Self-avoiding walks 16
1.8 Toda s theorems 17
1.9 Additional notes 19
2 Knots and links 20
2.1 Introduction 20
2.2 Tait colourings 23
2.3 Classifying knots 25
2.4 Braids and the braid group 29
2.5 The braid index and the Seifert graph of a link 31
2.6 Enzyme action 35
2.7 The number of knots and links 37
2.8 The topology of polymers 37
2.9 Additional notes 40
3 Colourings, flows and polynomials 42
3.1 The chromatic polynomial 42
3.2 The Whitney-Tutte polynomials 44
3.3 Tutte Grothendieck invariants 46
3.4 Reliability theory 48
3.5 Flows over an Abelian group 49
3.6 Ice models 50
3.7 A catalogue of invariants 51
3.8 Additional notes 53
4 Statistical physics 54
4.1 Percolation processes 54
4.2 The Ising model 58
4.3 Combinatorial interpretations 60
v
vi Contents
4.4 The Ashkin-Teller-Potts model 62
4.5 The random cluster model 64
4.6 Percolation in the random cluster model 67
4.7 Additional notes 70
5 Link polynomials and the Tait conjectures 71
5.1 The Alexander polynomial 71
5.2 The Jones polynomial and Kauffman bracket 75
5.3 The Homfly polynomial 82
5.4 The Kauffman 2-variable polynomial 84
5.5 The Tait conjectures 87
5.6 Thistlethwaite s nontriviality criterion 90
5.7 Link invariants and statistical mechanics 92
5.8 Additional notes 94
6 Complexity questions 95
6.1 Computations in knot theory 95
6.2 The complexity of the Tutte plane 96
6.3 The complexity of knot polynomials 100
6.4 The complexity of the Ising model 104
6.5 Reliability and other computations 107
6.6 Additional notes 109
7 The complexity of uniqueness and parity 110
7.1 Unique solutions 110
7.2 Unambiguous machines and one-way functions . 113
7.3 The Valiant-Vazirani theorem 114
7.4 Hard counting problems not parsimonious with SAT 117
7.5 The curiosity of parity 118
7.6 Toda s theorem on parity 122
7.7 Additional notes 123
8 Approximation and randomisation 124
8.1 Metropolis methods 124
8.2 Approximating to within a ratio 126
8.3 Generating solutions at random 129
8.4 Rapidly mixing Markov chains 130
8.5 Computing the volume of a convex body 132
8.6 Approximations and the Ising model 135
8.7 Some open questions 137
8.8 Additional notes 141
References 143
|
| any_adam_object | 1 |
| author | Welsh, Dominic J. A. |
| author_facet | Welsh, Dominic J. A. |
| author_role | aut |
| author_sort | Welsh, Dominic J. A. |
| author_variant | d j a w dja djaw |
| building | Verbundindex |
| bvnumber | BV008247478 |
| classification_rvk | SI 320 SK 300 SK 890 |
| classification_tum | MAT 057f |
| ctrlnum | (OCoLC)246644008 (DE-599)BVBBV008247478 |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV008247478 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:17:04Z |
| institution | BVB |
| isbn | 0521457408 |
| language | English |
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| physical | VIII, 163 S. graph. Darst. |
| publishDate | 1993 |
| publishDateSearch | 1993 |
| publishDateSort | 1993 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | London Mathematical Society: London Mathematical Society lecture note series |
| series2 | London Mathematical Society: London Mathematical Society lecture note series |
| spelling | Welsh, Dominic J. A. Verfasser aut Complexity knots, colourings and counting D. J. A. Welsh 1. publ. Cambridge u.a. Cambridge Univ. Press 1993 VIII, 163 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society: London Mathematical Society lecture note series 186 Kombinatorik (DE-588)4031824-2 gnd rswk-swf Knotentheorie (DE-588)4164318-5 gnd rswk-swf Abzählen (DE-588)4508960-7 gnd rswk-swf Komplexität (DE-588)4135369-9 gnd rswk-swf Statistische Physik (DE-588)4057000-9 gnd rswk-swf Enumeration (DE-588)4242271-1 gnd rswk-swf Knoten Mathematik (DE-588)4164314-8 gnd rswk-swf Abzählen (DE-588)4508960-7 s Komplexität (DE-588)4135369-9 s DE-604 Knoten Mathematik (DE-588)4164314-8 s Enumeration (DE-588)4242271-1 s Statistische Physik (DE-588)4057000-9 s Kombinatorik (DE-588)4031824-2 s Knotentheorie (DE-588)4164318-5 s 1\p DE-604 London Mathematical Society: London Mathematical Society lecture note series 186 (DE-604)BV000000130 186 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005446041&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Welsh, Dominic J. A. Complexity knots, colourings and counting London Mathematical Society: London Mathematical Society lecture note series Kombinatorik (DE-588)4031824-2 gnd Knotentheorie (DE-588)4164318-5 gnd Abzählen (DE-588)4508960-7 gnd Komplexität (DE-588)4135369-9 gnd Statistische Physik (DE-588)4057000-9 gnd Enumeration (DE-588)4242271-1 gnd Knoten Mathematik (DE-588)4164314-8 gnd |
| subject_GND | (DE-588)4031824-2 (DE-588)4164318-5 (DE-588)4508960-7 (DE-588)4135369-9 (DE-588)4057000-9 (DE-588)4242271-1 (DE-588)4164314-8 |
| title | Complexity knots, colourings and counting |
| title_auth | Complexity knots, colourings and counting |
| title_exact_search | Complexity knots, colourings and counting |
| title_full | Complexity knots, colourings and counting D. J. A. Welsh |
| title_fullStr | Complexity knots, colourings and counting D. J. A. Welsh |
| title_full_unstemmed | Complexity knots, colourings and counting D. J. A. Welsh |
| title_short | Complexity |
| title_sort | complexity knots colourings and counting |
| title_sub | knots, colourings and counting |
| topic | Kombinatorik (DE-588)4031824-2 gnd Knotentheorie (DE-588)4164318-5 gnd Abzählen (DE-588)4508960-7 gnd Komplexität (DE-588)4135369-9 gnd Statistische Physik (DE-588)4057000-9 gnd Enumeration (DE-588)4242271-1 gnd Knoten Mathematik (DE-588)4164314-8 gnd |
| topic_facet | Kombinatorik Knotentheorie Abzählen Komplexität Statistische Physik Enumeration Knoten Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005446041&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000130 |
| work_keys_str_mv | AT welshdominicja complexityknotscolouringsandcounting |