The concordance genus of knots

In knot concordance three genera arise naturally, g(K), g 4 (K), and g c (K): these are the classical genus, the 4-ball genus, and the concordance genus, defined to be the minimum genus among all knots concordant to K. Clearly 0 < g 4 (K) ≤ g c (K) ≤ g(K). Casson and Nakanishi gave examples to show that g 4 (K) need not equal g c (K). We begin by reviewing and extending their results. For knots representing elements in A, the concordance group of algebraically slice knots, the relationships between these genera are less clear. Casson and Gordon's result that A is nontrivial implies that g 4 (K) can be nonzero for knots in A. Gilmer proved that g 4 (K) can be arbitrarily large for knots in A. We will prove that there are knots K in A with g 4 (K) = 1 and g c (K) arbitrarily large. Finally, we tabulate g c for all prime knots with 10 crossings and, with two exceptions, all prime knots with fewer than 10 crossings. This requires the description of previously unnoticed concordances.

[1]  D. Rolfsen.  Knots and Links , 2003 .

[2]  渋谷 哲夫.  Local Moves and 4-Genus of Knots , 2000 .

[3]  C. Livingston,et al.  Concordance and mutation , 1999, math/9912174.

[4]  P. Teichner,et al.  Knot concordance, Whitney towers and L^2 signatures , 1999, math/9908117.

[5]  C. Livingston,et al.  TWISTED KNOT POLYNOMIALS: INVERSION, MUTATION AND CONCORDANCE , 1999 .

[6]  Charles Livingston,et al.  TWISTED ALEXANDER INVARIANTS, REIDEMEISTER TORSION, AND CASSON–GORDON INVARIANTS , 1999 .

[7]  E. Witten.  Monopoles and four-manifolds , 1994, hep-th/9411102.

[8]  Masaaki Wada,et al.  Twisted Alexander polynomial for finitely presentable groups , 1994 .

[9]  P. Kronheimer,et al.  Gauge theory for embedded surfaces, II , 1993 .

[10]  C. Livingston,et al.  The Casson-Gordon invariant and link concordance , 1992 .

[11]  L. Rudolph.  Some topologically locally-flat surfaces in the complex projective plane , 1984 .

[12]  P. Gilmer.  SLICE KNOTS IN S3 , 1983 .

[13]  P. Gilmer.  On the slice genus of knots , 1982 .

[14]  S. Akbulut,et al.  Branched covers of surfaces in 4-manifolds , 1980 .

[15]  Hisako Kondo.  Knots of unknotting number 1 and their Alexander polynomials , 1979 .

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

[17]  A. Tristram.  Some cobordism invariants for links , 1969, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  J. Levine.  Invariants of knot cobordism , 1969 .

[19]  K. Murasugi.  ON A CERTAIN NUMERICAL INVARIANT OF LINK TYPES , 1965 .

[20]  E. Artin.  Zur Isotopie zweidimensionaler Flächen imR4 , 1925 .

[21]  T. Kim.  On Knot Concordance , 2006 .

[22]  P. Kronheimer,et al.  Gauge theory for embedded surfaces , II , 1997 .

[23]  Akio Kawauchi,et al.  A Survey of Knot Theory , 1996 .

[24]  Michael H. Freedman,et al.  Topology of 4-manifolds , 1990 .

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

[26]  S. Donaldson.  An application of gauge theory to four-dimensional topology , 1983 .

[27]  Michael H. Freedman,et al.  The topology of four-dimensional manifolds , 1982 .

[28]  Yasutaka Nakanishi.  A Note on Unknotting Number , 1981 .

[29]  C. Gordon.  Some aspects of classical knot theory , 1978 .

[30]  T. Matumoto.  On the signature invariants of a non-singular complex sesqui-linear form , 1977 .

[31]  A. Casson,et al.  On slice knots in dimension three , 1976 .

[32]  M. Kervaire.  Knot cobordism in codimension two , 1971 .

[33]  J. Conway.  An enumeration of knots and links, and some of their algebraic properties , 1970 .

[34]  J. Milnor,et al.  Singularities of 2-spheres in 4-space and cobordism of knots , 1966 .

[35]  Horst Schubert,et al.  Die eindeutige Zerlegbarkeit eines Knotens in Primknoten , 1949 .

[36]  Robert Lipshitz,et al.  Algebraic & Geometric Topology , 2023 .