## The concordance genus of a knot. II.(English)Zbl 1171.57004

This paper is a sequel to the author’s paper [Algebr. Geom. Topol. 4, 1-22 (2004; Zbl 1055.57007)]. Let $$K$$ be a knot in the 3-sphere $$S^3$$. Let $$g_3(K)$$ be the minimum genus of an orientable embedded surface bounded by $$K$$ in $$S^3$$, and $$g_4(K)$$ the minimum genus of an orientable embedded surface bounded by $$K$$ in the 4-ball $$B^4$$. We denote by $$g_c(K)$$ the minimum value of $$g_3(J)$$ among all knots $$J$$ concordant to $$K$$. This paper extends previous techniques to compute these invariants, provides interesting examples and finishes the computations for knots with 10 or fewer crossings.
Concretely, the author proved:
Proposition 1. For $$K=8_{18}$$ and $$K=9_{40}$$, $$g_3(K)=g_c(K)=3$$ and $$g_4(K)=1$$. Proposition 2. For $$K=10_{82}, g_3(K)=4$$ and $$g_4(K)=1$$. There are knots $$J$$ with $$g_3(J)\leq 3$$ that are algebraically concordant to $$K$$, the first of which is $$9_{42}$$, but $$K$$ is not concordant to any such $$J$$. In particular, $$g_{c}(K)=4$$.
Further, the last section lists some related open problems.

### MSC:

 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57N70 Cobordism and concordance in topological manifolds

### Keywords:

knot genus; concordance; twisted Alexander polynomial

Zbl 1055.57007

KnotInfo
Full Text:

### References:

 [1] A J Casson, C M Gordon, Cobordism of classical knots (editors L Guillou, A Marin), Progr. Math. 62, Birkhäuser (1986) 181 [2] J C Cha, C Livingston, KnotInfo [3] T D Cochran, R E Gompf, Applications of Donaldson’s theorems to classical knot concordance, homology $$3$$-spheres and property $$P$$, Topology 27 (1988) 495 · Zbl 0669.57003 [4] T D Cochran, K E Orr, P Teichner, Knot concordance, Whitney towers and $$L^2$$-signatures, Ann. of Math. $$(2)$$ 157 (2003) 433 · Zbl 1044.57001 [5] D S Dummit, R M Foote, Abstract algebra, John Wiley & Sons (2004) · Zbl 1037.00003 [6] R H Fox, Free differential calculus. III. Subgroups, Ann. of Math. $$(2)$$ 64 (1956) 407 · Zbl 0073.25401 [7] R H Fox, J W Milnor, Singularities of $$2$$-spheres in $$4$$-space and cobordism of knots, Osaka J. Math. 3 (1966) 257 · Zbl 0146.45501 [8] C M Gordon, Some aspects of classical knot theory (editor J C Hausmann), Lecture Notes in Math. 685, Springer (1978) 1 · Zbl 0386.57002 [9] J E Grigsby, D Ruberman, S Strle, Knot concordance and Heegaard Floer homology invariants in branched covers, Geom. Topol. 12 (2008) 2249 · Zbl 1149.57007 [10] C Herald, P Kirk, C Livingston, Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation · Zbl 1210.57006 [11] S Jabuka, S Naik, Order in the concordance group and Heegaard Floer homology, Geom. Topol. 11 (2007) 979 · Zbl 1132.57008 [12] P Kirk, C Livingston, Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants, Topology 38 (1999) 635 · Zbl 0928.57005 [13] J Levine, Invariants of knot cobordism, Invent. Math. 8 $$(1969)$$ 98-110; addendum, ibid. 8 (1969) 355 · Zbl 0179.52401 [14] P Lisca, Sums of lens spaces bounding rational balls, Algebr. Geom. Topol. 7 (2007) 2141 · Zbl 1185.57015 [15] C Livingston, The algebraic concordance order of a knot · Zbl 1230.57008 [16] C Livingston, The concordance genus of knots, Algebr. Geom. Topol. 4 (2004) 1 · Zbl 1055.57007 [17] C Livingston, S Naik, Obstructing four-torsion in the classical knot concordance group, J. Differential Geom. 51 (1999) 1 · Zbl 1025.57013 [18] J Milnor, D Husemoller, Symmetric bilinear forms, Ergebnisse der Math. und ihrer Grenzgebiete 73, Springer (1973) · Zbl 0292.10016 [19] K Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965) 387 · Zbl 0137.17903 [20] Y Nakanishi, A note on unknotting number, Math. Sem. Notes Kobe Univ. 9 (1981) 99 · Zbl 0481.57002 [21] P Ozsváth, Z Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003) 615 · Zbl 1037.57027 [22] A Plans, Contribution to the study of the homology groups of the cyclic ramified coverings corresponding to a knot, Revista Acad. Ci. Madrid 47 (1953) 161 [23] J Rasmussen, Khovanov homology and the slice genus · Zbl 1211.57009 [24] D Rolfsen, Knots and links, Math. Lecture Ser. 7, Publish or Perish (1976) · Zbl 0339.55004 [25] W Scharlau, Quadratic and Hermitian forms, Grund. der Math. Wissenschaften 270, Springer (1985) · Zbl 0584.10010 [26] J P Serre, A course in arithmetic, Graduate Texts in Math. 7, Springer (1973) · Zbl 0256.12001 [27] N W Stoltzfus, Unraveling the integral knot concordance group, Mem. Amer. Math. Soc. 12 (1977) · Zbl 0366.57005 [28] A G Tristram, Some cobordism invariants for links, Proc. Cambridge Philos. Soc. 66 (1969) 251 · Zbl 0191.54703
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