Additive invariants for knots, links and graphs in 3–manifolds

We define two new families of invariants for (3-manifold, graph) pairs which detect the unknot and are additive under connected sum of pairs and (-1/2)-additive under trivalent vertex sum of pairs. The first of these families is closely related to both bridge number and tunnel number. The second of these families is a variation and generalization of Gabai's width for knots in the 3-sphere. We give applications to the tunnel number and higher genus bridge number of connected sums of knots.

[1]  Scott A. Taylor,et al.  Thin position for knots, links, and graphs in 3-manifolds , 2016, 1606.03331.

[2]  Tsuyoshi Kobayashi,et al.  The growth rate of the tunnel number of m-small knots , 2015, 1506.03916.

[3]  K. Morimoto.  On composite types of tunnel number two knots , 2014, 1409.0981.

[4]  Maggy Tomova,et al.  Width is not additive , 2010, 1005.1359.

[5]  V. Turaev,et al.  A semigroup of theta-curves in 3-manifolds , 2010, 1005.0308.

[6]  Maggy Tomova,et al.  COMPANIONS OF THE UNKNOT AND WIDTH ADDITIVITY , 2009, 0908.4103.

[7]  Marc Lackenby,et al.  An algorithm to determine the Heegaard genus of simple 3–manifolds with nonempty boundary , 2007, 0709.0376.

[8]  Maggy Tomova,et al.  Uniqueness of bridge surfaces for 2-bridge knots , 2006, Mathematical Proceedings of the Cambridge Philosophical Society.

[9]  Tsuyoshi Kobayashi,et al.  Heegaard genus of the connected sum of m-small knots , 2005, math/0503229.

[10]  M. Scharlemann,et al.  On the additivity of knot width , 2004, math/0403326.

[11]  J. Schultens.  Additivity of bridge numbers of knots , 2003, Mathematical Proceedings of the Cambridge Philosophical Society.

[12]  J. Schultens,et al.  3-manifolds with planar presentations and the width of satellite knots , 2003, math/0304271.

[13]  E. Sedgwick,et al.  Thin position for a connected sum of small knots , 2002, math/0205125.

[14]  Y. Moriah.  On boundary primitive manifolds and a theorem of Casson–Gordon , 2002, math/0201115.

[15]  C. Hayashi,et al.  Thin position of a pair (3-manifold, 1-submanifold) , 2001 .

[16]  Martin Scharlemann,et al.  Chapter 18 – Heegaard Splittings of Compact 3-Manifolds , 2000, math/0007144.

[17]  K. Morimoto.  Tunnel number, connected sum and meridional essential surfaces , 2000 .

[18]  J. Schultens,et al.  Tunnel numbers of small knots do not go down under connected sum , 1999 .

[19]  J. Schultens,et al.  THE TUNNEL NUMBER OF THE SUM OF n KNOTS IS AT LEAST n , 1999, math/9902008.

[20]  R. Weidmann.  On the rank of amalgamated products and product knot groups , 1998 .

[21]  S. Boyer.  Dehn surgery on knots , 1998 .

[22]  John Hempel,et al.  3-Manifolds as viewed from the curve complex ☆ , 1997, math/9712220.

[23]  M. Sakuma,et al.  Examples of tunnel number one knots which have the property ‘1 + 1 = 3’ , 1996, Mathematical Proceedings of the Cambridge Philosophical Society.

[24]  K. Morimoto.  There are knots whose tunnel numbers go down under connected sum , 1995 .

[25]  Yoshiyuki Yokota,et al.  On quantum SU(2) invariants and generalized bridge numbers of knots , 1995, Mathematical Proceedings of the Cambridge Philosophical Society.

[26]  J. Schultens.  The Classification of Heegaard Splittings for (Compact Orient Able Surface) × S1 , 1993 .

[27]  H. Doll.  A generalized bridge number for links in 3-manifolds , 1992 .

[28]  John Luecke,et al.  Knots are determined by their complements , 1989 .

[29]  Katura Miyazaki.  Conjugation and the prime decomposition of knots in closed, oriented 3-manifolds , 1989 .

[30]  C. McA. Gordon,et al.  Reducing heegaard splittings , 1987 .

[31]  David Gabai,et al.  Foliations and the topology of 3-manifolds , 1987 .

[32]  H. Schubert.  Über eine numerische Knoteninvariante , 1954 .

[33]  John Milnor,et al.  ON THE TOTAL CURVATURE OF KNOTS , 1950 .

[34]  David Gabai.  FOLIATIONS AND THE TOPOLOGY OF 3-MANIFOLDS. II , 2008 .

[35]  C. Hayashi,et al.  Heegaard splittings of the pair of the solid torus and the core loop , 2001 .

[36]  Abigail Thompson,et al.  THIN POSITION AND THE RECOGNITION PROBLEM FOR S3 , 1994 .

[37]  Abigail Thompson,et al.  Thin Position and the Recognition Problem for $\bold{S^3}$ , 1994 .

[38]  Martin Scharlemann,et al.  Thin position for 3-manifolds , 1992 .

[39]  F. Norwood.  Every two-generator knot is prime , 1982 .

[40]  Bradd Clark.  The Heegaard genus of manifolds obtained by surgery on links and knots , 1980 .

[41]  William Jaco,et al.  Lectures on three-manifold topology , 1980 .