Quandle coloring and cocycle invariants of composite knots and abelian extensions
Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle coloring of composite knots...
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World Scientific Publishing Co. Pte Ltd
2016
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| LEADER | 08514caa a22008417a 4500 | ||
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| 001 | PAPER-16074 | ||
| 003 | AR-BaUEN | ||
| 005 | 20230518204657.0 | ||
| 008 | 190411s2016 xx ||||fo|||| 00| 0 eng|d | ||
| 024 | 7 | |2 scopus |a 2-s2.0-84963692090 | |
| 040 | |a Scopus |b spa |c AR-BaUEN |d AR-BaUEN | ||
| 100 | 1 | |a Clark, W.E. | |
| 245 | 1 | 0 | |a Quandle coloring and cocycle invariants of composite knots and abelian extensions |
| 260 | |b World Scientific Publishing Co. Pte Ltd |c 2016 | ||
| 506 | |2 openaire |e Política editorial | ||
| 504 | |a Andruskiewitsch, N., Grana, M., From racks to pointed Hopf algebras (2003) Adv. Math., 178, pp. 177-243 | ||
| 504 | |a Birman, J.S., Menasco, W., Studying links via closed braids IV: Composite links and split links (1990) Invent. Math., 102, pp. 115-139 | ||
| 504 | |a Burde, G., Zieschang, H., (2003) Knots, de Gruyter Studies in Mathematics, 2nd Edn., 5. , Walter de Gruyter & Co., Berlin | ||
| 504 | |a Carter, J.S., Elhamdadi, M., Nikiforou, M.A., Saito, M., Extensions of quandles and cocycle knot invariants (2003) J. Knot Theory Ramifications, 12, pp. 725-738 | ||
| 504 | |a Carter, J.S., Jelsovsky, D., Kamada, S., Langford, L., Saito, M., Quandle cohomology and state-sum invariants of knotted curves and surfaces (2003) Trans. Amer. Math. Soc., 355, pp. 3947-3989 | ||
| 504 | |a Carter, J.S., Jelsovsky, D., Kamada, S., Saito, M., Computations of quandle cocycle invariants of knotted curves and surfaces (2001) Adv. Math., 157, pp. 36-94 | ||
| 504 | |a Carter, J.S., Jelsovsky, D., Kamada, S., Saito, M., Quandle homology groups, their Betti numbers, and virtual knots (2001) J Pure Appl. Algebra, 157 (2-3), pp. 135-155 | ||
| 504 | |a Carter, J.S., Kamada, S., Saito, M., Surfaces in 4-space (2004) Encyclopaedia of Mathematical Sciences, 142. , Springer Verlag | ||
| 504 | |a Carter, J.S., Kamada, S., Saito, M., Geometric interpretations of quandle homology (2001) J. Knot Theory Ramifications, 10, pp. 345-386 | ||
| 504 | |a Carter, J.S., Saito, M., Satoh, S., Ribbon concordance of surface-knots via quandle cocycle invariants (2006) J. Aust. Math. Soc., 80, pp. 131-147 | ||
| 504 | |a Clark, W.E., Elhamdadi, M., Saito, M., Yeatman, T., Quandle colorings of knots and applications (2014) J. Knot Theory Ramifications, 23, p. 1450035. , 29 | ||
| 504 | |a Clark, W.E., Yeatman, T., Properties of Rig Quandles, , http://math.usf.edu/?saito/QuandleColor/characterization.pdf | ||
| 504 | |a Conway, J.H., (1970) An Enumeration of Knots and Links, and Some of Their Algebraic Properties, in Computational Problems in Abstract Algebra, pp. 329-358. , ed. J. Leech ( Pergamon Press Oxford, England | ||
| 504 | |a Cha, J.C., Livingston, C., (2011) Knot Info: Table of Knot Invariants, , http://www.indiana.edu/?knotinfo, May 26 | ||
| 504 | |a Clauwens, F.J.B.J., Small Connected Quandles, , http://front.math.ucdavis.edu/1011.2456 | ||
| 504 | |a Eisermann, M., Homological characterization of the unknot (2003) J. Pure Appl. Algebra, 177 (2), pp. 131-157 | ||
| 504 | |a Elhamdadi, M., MacQuarrie, J., Restrepo, R., Automorphism groups of quandles (2012) J. Algebra Appl., 11, p. 1250008. , 9 | ||
| 504 | |a Fenn, R., Rourke, C., Racks and links in codimension two (1992) J. Knot Theory Ramifications, 1, pp. 343-406 | ||
| 504 | |a Fenn, R., Rourke, C., Sanderson, B., Trunks and classifying spaces (1995) Appl. Categ. Structures, 3, pp. 321-356 | ||
| 504 | |a Hempel, J., (1984) Residual Finiteness for 3-manifolds, Combinatorial Group Theory and Topology, pp. 379-396. , Alta, Utah | ||
| 504 | |a (1987) Ann. of Math. Stud., 111. , Princeton Univ. Press, Princeton, NJ | ||
| 504 | |a Joyce, D., A classifying invariant of knots, the knot quandle (1982) J Pure Appl. Algebra, 23, pp. 37-65 | ||
| 504 | |a Litherland, L.N., Nelson, S., The Betti numbers of some finite racks (2003) J. Pure Appl. Algebra, 178, pp. 187-202 | ||
| 504 | |a Manturov, V., (2004) Knot Theory, , Chapman & Hall/CRC, Boca Raton, FL | ||
| 504 | |a Matveev, S., Distributive groupoids in knot theory (1982) Mat. Sb. (N.S.) (Russian), 119 (161), pp. 78-88. , (160) | ||
| 504 | |a Mochizuki, T., The third cohomology groups of dihedral quandles (2011) J. Knot Theory Ramifications, 20, pp. 1041-1057 | ||
| 504 | |a Niebrzydowski, M., Przytycki, J.H., Homology of dihedral quandles (2009) J Pure Appl. Algebra, 213, pp. 742-755 | ||
| 504 | |a Nosaka, T., On homotopy groups of quandle spaces and the quandle homotopy invariant of links (2011) Topology Appl., 158, pp. 996-1011 | ||
| 504 | |a Papakyriakopoulos, C.D., On Dehn's lemma and the asphericity of knots (1957) Ann. Math., 66 (2), pp. 1-26 | ||
| 504 | |a Przytycki, J.H., (1995) 3-colorings and Other Elementary Invariants of Knots Knot Theory, pp. 275-295. , Warsaw | ||
| 504 | |a (1998) Polish Acad. Sci., Warsaw, 42. , Banach Center Publ | ||
| 504 | |a Przytycki, J.H., From 3-moves to Lagrangian tangles and cubic skein modules (2004) Advances in Topological Quantum Field Theory, Proceedings of the NATO ARW on New Techniques in Topological Quantum Field Theory, pp. 71-125. , ed John M. Bryden | ||
| 504 | |a Takasaki, M., Abstraction of symmetric transformation (1942) Tohoku Math. J. (In Japanese), 49, pp. 145-207 | ||
| 504 | |a Thurston, W.P., Three-dimensional manifolds, Kleinian groups and hyperbolic geometry (1982) Bull. Amer. Math. Soc. (N.S.), 6, pp. 357-381 | ||
| 504 | |a Vendramin, L., Rig-A GAP Package for Racks and Quandles, , http://github.com/vendramin/rig | ||
| 504 | |a Vendramin, L., On the classification of quandles of low order (2012) J. Knot Theory Ramifications, 21 (9), p. 1250088 | ||
| 520 | 3 | |a Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle coloring of composite knots. We investigate this and related phenomena. Quandle cocycle invariants are studied in relation to quandle coloring of the connected sum, and formulas are given for computing the cocycle invariant from the number of colorings of composite knots. Relations to corresponding abelian extensions of quandles are studied, and extensions are examined for the table of small connected quandles, called Rig quandles. Computer calculations are presented, and summaries of outputs are discussed. © 2016 World Scientific Publishing Company. |l eng | |
| 536 | |a Detalles de la financiación: National Institutes of Health, R01GM109459 | ||
| 536 | |a Detalles de la financiación: Abdus Salam International Centre for Theoretical Physics, UBACyT 20020110300037, PICT-2014-1376 | ||
| 536 | |a Detalles de la financiación: National Institutes of Health | ||
| 536 | |a Detalles de la financiación: MS was partially supported by the National Institutes of Health under Award Number R01GM109459. The content of this paper is solely the responsibility of the authors and does not necessarily represent the official views of NIH. LV was partially supported by Conicet, ICTP, UBACyT 20020110300037 and PICT-2014-1376. We thank Santiago Laplagne for the computer where some calculations were performed. | ||
| 593 | |a Department of Mathematics and Statistics, University of South Florida, Tampa, FL, United States | ||
| 593 | |a Departamento de Mathemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, Argentina | ||
| 690 | 1 | 0 | |a ABELIAN EXTENSIONS |
| 690 | 1 | 0 | |a COCYCLE INVARIANTS |
| 690 | 1 | 0 | |a COLORINGS |
| 690 | 1 | 0 | |a COMPOSITE KNOTS |
| 690 | 1 | 0 | |a QUANDLE |
| 700 | 1 | |a Saito, M. | |
| 700 | 1 | |a Vendramin, L. | |
| 773 | 0 | |d World Scientific Publishing Co. Pte Ltd, 2016 |g v. 25 |k n. 5 |p J. Knot Theory Ramifications |x 02182165 |w (AR-BaUEN)CENRE-5634 |t Journal of Knot Theory and its Ramifications | |
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