Homology groups of symmetric quandles and cocycle invariants of links and surface-links

Kamada, Seiichi; Oshiro, Kanako

doi:10.1090/S0002-9947-2010-05131-1

Homology groups of symmetric quandles and cocycle invariants of links and surface-links
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Trans. Amer. Math. Soc. 362 (2010), 5501-5527 Request permission

Abstract:

We introduce the notion of a quandle with a good involution and its homology groups. Carter et al. defined quandle cocycle invariants for oriented links and oriented surface-links. By use of good involutions, quandle cocyle invariants can be defined for links and surface-links which are not necessarily oriented or orientable. The invariants can be used in order to estimate the minimal triple point numbers of non-orientable surface-links.

References

Nicolás Andruskiewitsch and Matías Graña, From racks to pointed Hopf algebras, Adv. Math. 178 (2003), no. 2, 177–243. MR 1994219, DOI 10.1016/S0001-8708(02)00071-3
Soichiro Asami and Shin Satoh, An infinite family of non-invertible surfaces in 4-space, Bull. London Math. Soc. 37 (2005), no. 2, 285–296. MR 2119028, DOI 10.1112/S0024609304003832
J. Scott Carter, Mohamed Elhamdadi, Matias Graña, and Masahico Saito, Cocycle knot invariants from quandle modules and generalized quandle homology, Osaka J. Math. 42 (2005), no. 3, 499–541. MR 2166720
J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, Laurel Langford, and Masahico Saito, Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc. 355 (2003), no. 10, 3947–3989. MR 1990571, DOI 10.1090/S0002-9947-03-03046-0
J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, and Masahico Saito, Quandle homology groups, their Betti numbers, and virtual knots, J. Pure Appl. Algebra 157 (2001), no. 2-3, 135–155. MR 1812049, DOI 10.1016/S0022-4049(00)00013-X
J. Scott Carter, Seiichi Kamada, and Masahico Saito, Geometric interpretations of quandle homology, J. Knot Theory Ramifications 10 (2001), no. 3, 345–386. MR 1825963, DOI 10.1142/S0218216501000901
J. Scott Carter, Seiichi Kamada, Masahico Saito, and Shin Satoh, A theorem of Sanderson on link bordisms in dimension 4, Algebr. Geom. Topol. 1 (2001), 299–310. MR 1834778, DOI 10.2140/agt.2001.1.299
J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, Mathematical Surveys and Monographs, vol. 55, American Mathematical Society, Providence, RI, 1998. MR 1487374, DOI 10.1090/surv/055
Roger Fenn and Colin Rourke, Racks and links in codimension two, J. Knot Theory Ramifications 1 (1992), no. 4, 343–406. MR 1194995, DOI 10.1142/S0218216592000203
Roger Fenn, Colin Rourke, and Brian Sanderson, Trunks and classifying spaces, Appl. Categ. Structures 3 (1995), no. 4, 321–356. MR 1364012, DOI 10.1007/BF00872903
Roger Fenn, Colin Rourke, and Brian Sanderson, The rack space, Trans. Amer. Math. Soc. 359 (2007), no. 2, 701–740. MR 2255194, DOI 10.1090/S0002-9947-06-03912-2
Masahide Iwakiri, Calculation of dihedral quandle cocycle invariants of twist spun 2-bridge knots, J. Knot Theory Ramifications 14 (2005), no. 2, 217–229. MR 2128511, DOI 10.1142/S0218216505003798
David Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982), no. 1, 37–65. MR 638121, DOI 10.1016/0022-4049(82)90077-9
Seiichi Kamada, Wirtinger presentations for higher-dimensional manifold knots obtained from diagrams, Fund. Math. 168 (2001), no. 2, 105–112. MR 1852735, DOI 10.4064/fm168-2-1
Seiichi Kamada, Knot invariants derived from quandles and racks, Invariants of knots and 3-manifolds (Kyoto, 2001) Geom. Topol. Monogr., vol. 4, Geom. Topol. Publ., Coventry, 2002, pp. 103–117. MR 2002606, DOI 10.2140/gtm.2002.4.103
Seiichi Kamada, Quandles with good involutions, their homologies and knot invariants, Intelligence of low dimensional topology 2006, Ser. Knots Everything, vol. 40, World Sci. Publ., Hackensack, NJ, 2007, pp. 101–108. MR 2371714, DOI 10.1142/9789812770967_{0}013
R. A. Litherland and Sam Nelson, The Betti numbers of some finite racks, J. Pure Appl. Algebra 178 (2003), no. 2, 187–202. MR 1952425, DOI 10.1016/S0022-4049(02)00211-6
S. V. Matveev, Distributive groupoids in knot theory, Mat. Sb. (N.S.) 119(161) (1982), no. 1, 78–88, 160 (Russian). MR 672410
Takuro Mochizuki, Some calculations of cohomology groups of finite Alexander quandles, J. Pure Appl. Algebra 179 (2003), no. 3, 287–330. MR 1960136, DOI 10.1016/S0022-4049(02)00323-7
Dennis Roseman, Reidemeister-type moves for surfaces in four-dimensional space, Knot theory (Warsaw, 1995) Banach Center Publ., vol. 42, Polish Acad. Sci. Inst. Math., Warsaw, 1998, pp. 347–380. MR 1634466
C. Rourke and B. Sanderson, There are two $2$-twist spun trefoils, preprint, arxiv:math.GT/0006062:v1.
C. Rourke and B. Sanderson, A new classification of links and some calculation using it, preprint, arxiv:math.GT/0006062:v2.
K. Oshiro, Homology groups of trivial quandles with good involutions and triple linking numbers of surface-links, J. Knot Theory Ramifications, to appear.
Shin Satoh, Triple point invariants of non-orientable surface-links, Proceedings of the First Joint Japan-Mexico Meeting in Topology (Morelia, 1999), 2002, pp. 207–218. MR 1903691, DOI 10.1016/S0166-8641(01)00118-3
S. Satoh, A note on the shadow cocycle invariant of a knot with a base point, J. Knot Theory Ramifications 16 (2007), no. 7, 959–967. MR 2354269, DOI 10.1142/S0218216507005579
Shin Satoh and Akiko Shima, The 2-twist-spun trefoil has the triple point number four, Trans. Amer. Math. Soc. 356 (2004), no. 3, 1007–1024. MR 1984465, DOI 10.1090/S0002-9947-03-03181-7
Shin Satoh and Akiko Shima, Triple point numbers and quandle cocycle invariants of knotted surfaces in 4-space, New Zealand J. Math. 34 (2005), no. 1, 71–79. MR 2141479
Mituhisa Takasaki, Abstraction of symmetric transformations, Tôhoku Math. J. 49 (1943), 145–207 (Japanese). MR 21002