Local surgery formulas for quantum invariants and the Arf invariant

A formula for the Arf invariant of a link is given in terms of the singularities of an immersed surface bounded by the link. This is applied to study the computational complexity of quantum invariants of 3{manifolds. AMS Classication 57M27; 68Q15

[1]  N. Sato.  Cobordisms of semi-boundary links , 1984 .

[2]  Richard Porter,et al.  Milnor’s -invariants and Massey products , 1980 .

[3]  T. Ohtsuki.  A polynomial invariant of rational homology 3-spheres , 1996 .

[4]  D. Sullivan,et al.  On the Kervaire Obstruction , 1971 .

[5]  Raymond A. Robertello.  An invariant of knot cobordism , 1965 .

[6]  Quantum cyclotomic orders of 3-manifolds☆ , 1998, math/9809129.

[7]  R. Kirby.  The topology of 4-manifolds , 1989 .

[8]  J. Hoste.  A formula for Casson’s invariant , 1986 .

[9]  Edward Witten,et al.  Quantum field theory and the Jones polynomial , 1989 .

[10]  V. Turaev.  Milnor invariants and Massey products , 1979 .

[11]  Vladimir Turaev,et al.  Invariants of 3-manifolds via link polynomials and quantum groups , 1991 .

[12]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[13]  J. Hoste.  The Arf invariant of a totally proper link , 1984 .

[14]  Ulrich Pinkall,et al.  Regular homotopy classes of immersed surfaces , 1985 .

[15]  Paul Melvin,et al.  The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2, C) , 1991 .

[16]  S. J. Kaplan.  Constructing framed 4-manifolds with given almost framed boundaries , 1979 .

[17]  V. Rokhlin,et al.  Topology and geometry : Rohlin seminar , 1988 .

[18]  B. Mellor,et al.  A Geometric Interpretation of Milnor's Triple Invariants , 2001 .

[19]  Louis H. Kauffman.  The conway polynomial , 1981 .

[20]  Edgar H. Brown,et al.  Generalizations of the Kervaire Invariant , 1972 .

[21]  K. Murasugi.  On the Arf invariant of links , 1984, Mathematical Proceedings of the Cambridge Philosophical Society.

[22]  Patrick M. Gilmer,et al.  A Method for Computing the Arf invariants of links , 1993 .

[23]  Tim D. Cochran.  Concordance invariance of coefficients of Conway's link polynomial , 1985 .

[24]  J. Levine.  Knot cobordism groups in codimension two , 1969 .

[25]  S. Donaldson,et al.  Pin structures on low-dimensional manifolds , 1991 .

[26]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[27]  M. Saito.  ON THE UNORIENTED SATO-LEVINE INVARIANT , 1993 .

[28]  A. Marin,et al.  A la recherche de la topologie perdue , 1986 .