Quandle Cocycle Invariants for Spatial Graphs and Knotted Handlebodies

Ishii, Atsushi ; Iwakiri, Masahide

Canadian journal of mathematics, Tome 64 (2012) no. 1, pp. 102-122

Résumé

We introduce a flow of a spatial graph and see how invariants for spatial graphs and handlebody-links are derived from those for flowed spatial graphs. We define a new quandle (co)homology by introducing a subcomplex of the rack chain complex. Then we define quandle colorings and quandle cocycle invariants for spatial graphs and handlebody-links.

Détail

DOI : 10.4153/CJM-2011-035-0

Mots-clés : 57M27, 57M15, 57M25, quandle cocycle invariant, knotted handlebody, spatial graph

Bibliographie

[1] [1] Carter, J. S., Jelsovsky, D., Kamada, S., Langford, L., and Saito, M., Quandle cohomology and state-sum invariants of knotted curves and surfaces. Trans. Amer. Math. Soc. 355(2003), no. 10, 3947-3989. http://dx. doi. org/10.1090/S0002-9947-03-03046-0 Google Scholar

[2] [2] Carter, J. S., Kamada, S., and Saito, M., Geometric interpretations of quandle homology. J. Knot Theory Ramifications 10(2001), no. 3, 345-386. http://dx. doi. org/10.1142/S0218216501000901 Google Scholar

[3] [3] Ishii, A., Moves and invariants for knotted handlebodies. Algebr. Geom. Topol. 8(2008), no. 3, 1403-1418. http://dx. doi. org/10.2140/agt.2008.8.1403 Google Scholar

[4] [4] Iwakiri, M., Triple point cancelling numbers of surface links and quandle cocycle invariants. Topology Appl. 153(2006), no. 15, 2815-2822. http://dx. doi. org/10.1016/j. topol.2005.12.001 Google Scholar

[5] [5] Iwakiri, M., The lower bound of the w-indices of surface links via quandle cocycle invariants. Trans. Amer. Math. Soc. 362(2010), no. 3, 1189-1210. http://dx. doi. org/10.1090/S0002-9947-09-04769-2 Google Scholar

[6] [6] Joyce, D., A classifying invariant of knots, the knot quandle. J. Pure Appl. Alg. 23(1982), no. 1, 37-65. http://dx. doi. org/10.1016/0022-4049(82)90077-9 Google Scholar

[7] [7] Joyce, D., Simple quandles. Algebra, J. 79(1982), no. 2, 307-318. http://dx. doi. org/10.1016/0021-8693(82)90305-2 Google Scholar

[8] [8] Kauffman, L. H., Invariants of graphs in three-space. Trans. Amer. Math. Soc. 311(1989), no. 2, 697-710. http://dx. doi. org/10.1090/S0002-9947-1989-0946218-0 Google Scholar

[9] [9] Matveev, S. V., Distributive groupoids in knot theory. Mat. Sb. (N. S.) 119(161)(1982), no. 1, 78-88, 160. Google Scholar

[10] [10] Mochizuki, T., Some calculations of cohomology groups of finite Alexander quandles. J. Pure Appl. Algebra 179(2003), no. 3, 287-330. http://dx. doi. org/10.1016/S0022-4049(02)00323-7 Google Scholar

[11] [11] Mochizuki, T., The 3-cocycles of the Alexander quandles Fq [T]/(T — !). Algebr. Geom. Topol. 5(2005), 183-205. http://dx. doi. org/10.2140/agt.2005.5.183 Google Scholar

[12] [12] Rourke, C. and Sanderson, B., There are two 2-twist-spun trefoils. http://citeseerx. ist. psu. edu/viewdoc/summary?doi=10.1.1.65.3250 Google Scholar

[13] [13] Satoh, S. and Shima, A., The 2-twist-spun trefoil has the triple point number four. Trans. Amer. Math. Soc. 356(2004), no. 3, 1007-1024. http://dx. doi. org/10.1090/S0002-9947-03-03181-7 Google Scholar

[14] [14] Suzuki, S., On linear graphs in 3-sphere. Osaka J. Math. 7(1970), 375-396. Google Scholar

[15] [15] Takasaki, M., Abstraction of symmetric transformations. (Japanese) Tohoku Math. J. 49(1943), 145-207. Google Scholar

[16] [16] Yamada, S., An invariant of spatial graphs. J. Graph Theory 13(1989), no. 5, 537-551. http://dx. doi. org/10.1002/jgt.3190130503 Google Scholar

[17] [17] Yetter, D. N., Category theoretic representations of knotted graphs in S3. Adv. Math. 77(1989), no. 2, 137-155. http://dx. doi. org/10.1016/0001-8708(89)90017-0 Google Scholar