Amphicheirality of links and Alexander invariants
Tóm tắt
We obtain an equation among invariants obtained from the Alexander module of an amphicheiral link. For special cases, it deduces necessary conditions on the Alexander polynomial. By using the present results and some known results, we show that the Alexander polynomial of an algebraically split component-preservingly (±)-amphicheiral link with even components is zero, and we determine prime amphicheiral links with at least 2 components and up to 9 crossings.
Tài liệu tham khảo
Bar-Natan D, Morrison S, Atlas K. http://katlas.math.toronto.edu/wiki/Main_Page
Bonahon F, Siebenmann L C. New geometric splittings of classical knots and the classification and symmetries of arborescent knots. Preprint
Hartley R. Invertible amphicheiral knots. Math Ann, 1980, 252: 103–109
Hartley R, Kawauchi A. Polynomials of amphicheiral knots. Math Ann, 1979, 243: 63–70
Hillman J. Symmetries of knots and links, and invariants of abelian coverings (Part I). Kobe J Math, 1986, 3: 7–27
Jiang B, Lin X, Wang S C, et al. Achirality of knots and links. Topology Appl, 2002, 119: 185–208
Kadokami T. The link-symmetric groups of 2-bridge links. J Knot Theory Ramifications, in press
Kadokami T. Amphicheiral links with special properties, I. J Knot Theory Ramifications, in press
Kadokami T. Amphicheiral links with special properties, II. J Knot Theory Ramifications, in press
Kawauchi A. On quadratic forms of 3-manifolds. Invent Math, 1977, 43: 177–198
Kawauchi A. The invertibility problem on amphicheiral excellent knots. Proc Japan Acad Ser A Math Sci, 1979, 55: 399–402
Kawauchi A. On the Rochlin invariants of Z 2-homology 3-spheres with cyclic actions. Japan J Math (N S), 1982, 8: 217–258
Kawauchi A. A survey of Knot Theory. Boston: Birkhäuser Verlag, 1996
Kawauchi A. The quadratic form of a link. Contemp Math, 1999, 233: 97–116
Kojima S, Yamasaki M. Some new invariants of links. Invent Math, 1979, 54: 213–228
Matsumoto Y, Venema G. Failure of the Dehn Lemma on contractible 4-manifolds. Invent Math, 1979, 51: 205–218
Milnor JW. Infinite cyclic coverings. In: Conference on the Topology of Manifolds. Boston: Prindle, Weber & Schmidt, 1968, 115–133
Murasugi K. On the signature of links. Topology, 1970, 9: 283–298
Nikkuni R, Taniyama K. Symmetries of spatial graphs and Simon invariants. Fund Math, 2009, 205: 219–236
Rolfsen D. Knots, Links. Berkeley: Publish or Perish Inc, 1976
Sakai T. Polynomials of invertible knots. Math Ann, 1983, 266: 229–232
Torres G. On the Alexander polynomial. Ann of Math (2), 1953, 57: 57–89
Traldi L. Milnor’s invariants and the completions of link modules. Trans Amer Math Soc, 1984, 284: 401–424
Turaev V. Reidemeister torsion in knot theory. Russian Math Surveys, 1986, 41: 119–182
Wang S D. Strict achirality of prime links up to 11-crossings. Acta Math Sin Engl Ser, 2008, 24: 997–1004
Whitten W. Symmetries of links. Trans Amer Math Soc, 1969, 135: 213–222