Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-22T17:44:17.267Z Has data issue: false hasContentIssue false
  • English
  • Français

Quandle Cocycle Invariants for Spatial Graphs and Knotted Handlebodies

Published online by Cambridge University Press:  20 November 2018

Atsushi Ishii  and
Masahide Iwakiri
Atsushi Ishii
Affiliation:
Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan email: [email protected]
Masahide Iwakiri
Affiliation:
Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

57M2757M1557M25quandle cocycle invariantknotted handlebodyspatial graph
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 1 , 01 February 2012 , pp. 102 - 122
Copyright
Copyright © Canadian Mathematical Society 2012

References

[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/S0002-9947-03-03046-0Google Scholar
[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/S0218216501000901Google Scholar
[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.2008.8.1403Google Scholar
[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.2005.12.001Google Scholar
[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/S0002-9947-09-04769-2Google Scholar
[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)90077-9Google Scholar
[7] Joyce, D., Simple quandles. Algebra, J. 79(1982), no. 2, 307-318. http://dx. doi. org/10.1016/0021-8693(82)90305-2Google Scholar
[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/S0002-9947-1989-0946218-0Google Scholar
[9] Matveev, S. V., Distributive groupoids in knot theory. Mat. Sb. (N. S.) 119(161)(1982), no. 1, 78-88, 160.Google Scholar
[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)00323-7Google Scholar
[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.2005.5.183Google Scholar
[12] Rourke, C. and Sanderson, B., There are two 2-twist-spun trefoils. http://citeseerx. ist. psu. edu/viewdoc/summary?doi=10.1.1.65.3250Google Scholar
[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/S0002-9947-03-03181-7Google Scholar
[14] Suzuki, S., On linear graphs in 3-sphere. Osaka J. Math. 7(1970), 375-396.Google Scholar
[15] Takasaki, M., Abstraction of symmetric transformations. (Japanese) Tohoku Math. J. 49(1943), 145-207.Google Scholar
[16] Yamada, S., An invariant of spatial graphs. J. Graph Theory 13(1989), no. 5, 537-551. http://dx. doi. org/10.1002/jgt.3190130503Google Scholar
[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)90017-0Google Scholar