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J. Scott Carter

J. Scott Carter

Professor Emeritus

Department of Math and Statistics

Education

B.S. March 1978, Summa Cum Laude,
University of Georgia, Athens, Georgia,
Mathematics Major
Ph.D. in Mathematics, May 1982
Yale University, New Haven, Connecticut
Dissertation: "Surgery on Immersions: a Geometric Approach to Stable Homotopy," Advisor Prof. Ronnie Lee.

Publications

J. Scott Carter Published Articles ...

with Seung Yeop Yang, Twist spinning knotted trivalent graphs
with Yongju Bae, Seonmi Choi, and Sera Kim, Non-orientable surfaces in 4-dimensional space
with Seiichi Kamada, Three dimensional braids and their descriptions, special issue of Topology and Its Applications, International Conference on Topology and Geometry 2013, the Sixth Japan-Mexico Topology
with Daniel Silver and Susan Williams. Invariants of links in thickened surfaces, Algebr. Geom. Topol. 14 (2014), no. 3, 1377 1394.
with Daniel Silver and Susan Williams. Three dimensions of knot coloring, Amer. Math. Monthly 121 (2014), no. 6, 506 514.
with Seiichi Kamada : How to fold a manifold, to appear in New Ideas in Low Dimensional Topology, ed. Kauffman and Manturov.

Books

Diagrammatic Algebra, with Seiichi Kamada, Mathematical Surveys and Monographs Volume: 264; American Mathematical Society (Providence 2021)
How Surfaces Intersect in Space: an Introduction to Topology, World Scientific Publishing, Singapore (First edition 1993), (Second edition 1995)
The Classical and Quantum 6j-symbols with D.E. Flath and M. Saito, Princeton University Press (1995).
Knotted Surfaces and Their Diagrams, with Masahico Saito American Mathematical Society.
Surfaces in 4-space, with Masahico Saito and Seiichi Kamada, Springer-Verlag.
Intelligence of Low Dimensional Topology, with Seiichi Kamada, Lou Kauffman, Akio Kawauchi, and Toshitake Kohno
An Excursion in Diagrammatic Algebra: turning a sphere from red to blue, World Science Publishing, Series on Knots and Everything, Volume 48.

Link to Additional Article Preprints

More articles published by J. Scott Carter

Reidemeister/Roseman-type Moves to Embedded Foams in 4-dimensional Space, to appear in New Ideas in Low Dimensional Topology, ed. Kauffman and Manturov.
with Atsushi Ishii. A knotted 2-dimensional foam with non-trivial cocycle invariant. Published in unrefereed conference Proceedings, Intelligence of Low Dimensional Topology, RIMS.
with Seiichi Kamada: Braids and Branched Coverings of Dimension Three, Published in unrefereed conference Proceedings, Intelligence of Low Dimensional Topology, RIMS, with Seiichi Kamada.
Classical Knot Theory, Symmetry (2012), 4, 225-250.
A Survey of Quandle Ideas, preprint, to appear in conference proceedings of Trieste form May 2009.
with Masahico Saito, Frobenius Modules and Essential Surface Cobordisms, preprint.
with Masahico Saito, Algebraic Structures Derived from Foams, to appear J. Gen. Lie Theory Appl.
with Masahico Saito and Kanako Oshiro, Symmetric extensions of dihedral quandles and triple points of non-orientable surfaces. Topology Appl. 157 (2010), no. 5, 857-869.
with Abhijit Champanerkar, A geometric method to compute some elementary integrals, preprint.
with Alissa Crans, Mohamed Elhamdadi, Masahico Saito, Cohomology of Algebraic Identities and R-Matrices, preprint.
with Alissa Crans, Mohamed Elhamdadi, Masahico Saito, Cohomology of Categorical Self-Distributivity, Journal of Homotopy Theory and Related Structures 3 (2008), no. 1, 13-63.
with Alissa Crans, Mohamed Elhamdadi, Masahico Saito, Cohomology of the Adjoint Map of Hopf Algebras. J. Gen. Lie Theory Appl. 2 (2008), no. 1, 19--34.
with Masahico Saito, Cohomology for self-distributivity in coalgebras. Intelligence of low dimensional topology 2006, 9--17, Ser. Knots Everything, 40, World Sci. Publ., Hackensack, NJ, 2007.
with Masahico Saito, Shin Satoh, Ribbon-moves for 2-knots with 1-handles attached and Khovanov-Jacobson numbers. Proc. Amer. Math. Soc. 134 (2006), no. 9, 2779--2783.
with Masahico Saito, Set Theoretic Yang-Baxter Solutions via Fox Calculus. J. Knot Theory Ramifications 15 (2006), no. 8, 949--956.
with Mohamed Elhamdadi, Shin Satoh and Masahico Saito, A lower bound for the number fo Reidemeister moves of type III, Topology Appl. 153 (2006), no. 15, 2788--2794.
with Shin Satoh and Masahico Saito, Ribbon concordance of surface knots via quandle cocycle invariants, J. Aust. Math. Soc. 80 (2006), no. 1, 131--147.
with Mohamed Elhamdadi, Matias Grana, and Masahico Saito, Cocycle knot invariants from quandle modules and generalized quandle homology. Osaka J. Math. 42 (2005), no. 3 499--541.
with Mohamed Elhamdadi and Masahico Saito, Homology theory for the set-theoretic Yang-Baxter equation and knot invariants from generalizations of quandles. Fund. Math. 184 (2004), 31--54.
with Masahico Saito. Quandle homology theory and cocycle knot invariants. Topology and geometry of manifolds (Athens, GA, 2001), 249--268, Proc. Sympos. Pure Math., 71, Amer. Math. Soc., Providence, RI, 2003.
with Masahico Saito and Seiichi Kamada, Diagrammatic computations for quandles and cocycle knot invariants. Diagrammatic morphisms and applications (San Francisco, CA, 2000), 51--74, Contemp. Math., 318, Amer. Math. Soc., Providence, RI, 2003.
with Mohamed Elhamdadi, Angela Harris, Marina Nikiforou and Masahico Saito, Cocycle knot invariants, quandle extensions, and Alexander matrices. Low-dimensional topology of tomorrow (Japanese) (Kyoto, 2002).
with Shin Satoh, Masahico Saito and Seiichi Kamada, Bordism of unoriented surfaces in 4-space. Michigan Math. J. 50 (2002), no. 3, 575--591.
with Masahico Saito and Seiichi Kamada, Stable equivalence of knots on surfaces and virtual knot cobordisms. Knots 2000 Korea, Vol. 1 (Yongpyong). J. Knot Theory Ramifications 11 (2002), no. 3, 311--322.
with Mohamed Elhamdadi, and Masahico Saito, Twisted quandle homology theory and cocycle knot invariants. Algebr. Geom. Topol. 2 (2002), 95--135.
with Shin Satoh, Masahico Saito and Seiichi Kamada, A theorem of Sanderson on link bordisms in dimension 4, Algebr. Geom. Topol. 1 (2001), 299--310.
with Daniel Jelsovsky, Masahico Saito and Seiichi Kamada, Shifting homomorphisms in quandle cohomology and skeins of cocycle knot invariants. J. Knot Theory Ramifications 10 (2001), no. 4, 579--596.
with Daniel Jelsovsky, Masahico Saito and Seiichi Kamada, Geometric interpretations of quandle homology. J. Knot Theory Ramifications 10 (2001), no. 3, 345--386.
with Daniel Jelsovsky, Masahico Saito and Seiichi Kamada, Quandle Homology Groups, Their Betti Numbers, and Virtual Knots. J. Pure Appl. Algebra 157 (2001), no. 2-3, 135--155.
with Daniel Jelsovsky, Masahico Saito and Seiichi Kamada, Computations of quandle cocycle invariants of knotted curves and surfaces. Adv. Math. 157 (2001), no. 1, 36--94.
with Daniel Jelsovsky, Masahico Saito, Seiichi Kamada, and Laurel Langford, Quandle cohomology and state-sum invariants of knotted curves and surfaces. Trans. Amer. Math. Soc. 355 (2003), no. 10, 3947--3989.
with Louis H. Kauffman and Masahico Saito, Structures and Diagrammatics of 4-Dimensional Topological Lattice Field Theories, Advances in Math., 146, 39-100 (1999).
with Masahico Saito and Seiichi Kamada, Alexander numbering of knotted surface diagrams, Proc. Amer. Math. Soc., 128 (2000), no. 12, 3761--3771.
with Masahico Saito and Vera Carrara, Singularities of Surface Maps and Knots, Pacific J. Math. 199 (2001), no. 1, 21--40.
with Sostenes Lins, Thin-G theory and local moves for gems, Adv. Math. 143 (1999), no. 2, 251--283. Advances in Mathematics, 127, No. 1, April 15 (1997), 1-51.
with Louis H. Kauffman and Masahico Saito, Diagrammatics, Singularities, and Their Algebraic Interpretations, 10th Brazilian Topology Meeting (São Carlos, 1996). Mat. Contemp. 13 (1997), 21--115.
with Masahico Saito, Surfaces in 3-Space that Do Not Lift to Embeddings in 4-Space, to appear in Conference Proceedings of the 1995 Warsaw Workshop on Knot Theory.
with Masahico Saito, and Joachim Rieger, A Combinatorial Description of Knotted Surfaces and Their Isotopies, Advances in Mathematics, 127, No. 1, April 15 (1997), 1-51.
with Masahico Saito, Normal Euler Classes of Knotted Surfaces and Triple Points on Their Projections, Proc. Amer. Math. Soc., 125 (1997), no. 2, 617-623.
with Masahico Saito, On Formulations and Solutions of Simplex Equations, International Journal of Modern Physics A, Vol 11, No 24, (1996), 4453-4463.
with Masahico Saito, Braid and Movies, Journal of Knot Theory and Its Ramifications, Vol 5, No. 5 (1996), 589-608.
with Masahico Saito, A Seifert Algorithm for Knotted Surfaces, Topology, Vol 36, No. 1 (1996), 179-201.
with Masahico Saito, Knot Diagrams and Braid Theories in Dimension 4 Real and Complex Singularities, ed. W. L Marar, Pitman Research Notes in Mathematics, Longman Publishing (1995).
with Masahico Saito, New Solutions to the Permutohedron Equation in ``Quantum Topology Kansas 1993,'' ed. David Yetter, World Science Publishing Company (1994), 51-65.
with Masahico Saito, Knotted Surfaces, Braid Movies, and Beyond, in ``Quantum Gravity,'' ed. John Baez, Oxford University Press (1994), 191-229.
with Masahico Saito, Reidemeister Moves for Surface Isotopies and Their Interpretations As Moves to Movies, Journal of Knot Theory and its Ramifications Vol 2, No 3 (1993), 251-284.
Solving Inequalities, Alabama Journal of Mathematics, Vol 1, No.1 \& 2, (1992), 31-35.
with Masahico Saito, A Diagrammatic Theory of Knotted Surfaces, in ``Quantum Topology,'' ed. Randy Baadhio and Louis Kauffman, World Science Publishing (Singapore 1993), 91-115.
with Masahico Saito, Planar Generalizations of the Yang Baxter Equation and Their Skeins, Journal of Knot Theory and its Ramifications Vol 1, No. 2 (1992), 207-217.
with Masahico Saito, Syzygies among Elementary String Interactions in Dimension 2+1, Letters in Mathematical Physics 23 (1991), 287-300.
Extending Immersed Circles in the Sphere to Immersed disks in the Ball, Comm. Math. Helv. 67 (1992), 337-348.
with Masahico Saito, Canceling Branch Points on Projections of Surfaces in 4-Space, Proc. of the AMS. 116, No 1. (Sept 1992), 229-237.
Closed Curves that Never Extend to Proper Maps of Disks, Proc. of the AMS 113, No. 3 (Nov 1991), 879-888.
Extending Immersed Curves to Proper Immersions of Surface, Topology and its Applications 40 (1991), 287-306.
Classifying Immersed Curves, Proc. of the AMS 111, No.1 (Jan. 1991), 281-287.
Immersed Projective Planes in Lens Spaces, Proc. of the AMS 106, No. 1 (May 1989), 251-260.
Immersed Codimension One Projective Spaces in Spherical Space Forms, Proc. of the AMS 105, No. 1, January 1989, 254-257.
with Ki Hyoung Ko, Triple Points of Immersed Surfaces in Three Dimensional Manifolds, Topology and Its Applications 32, (1989), 149-159.
Surgery on the Equatorial Immersion in Low Dimensions, Differential Topology Proceedings, Siegen 1987, Springer LMN 1350, U. Koschorke, ed.
Surgery on the Equatorial Immersion I, Illinois Journal of Mathematics 32, No. 4, Winter 1988, 703-715.
Simplifying the Self Intersection Sets of Codimension One Immersions in (n+1)-space, Houston Journal of Mathematics 13, No. 3, 353-366.
A Further Generalization of Boy's Surface, Houston Journal of Mathematics 12, No. 1 (1986), 11-31.
On Generalizing Boy's Surface: Constructing a Generator of the Third Stable Stem, Trans. of the AMS 289, No. 1 (1986), 103-121.
Surgery on Codimension One Immersions in (n+1)-space: Removing n-tuple Points, Trans. of the AMS 298, No. 1, (1986), 83-102.
Surgery Theory of Immersions, Proc. Northwestern Homotopy Theory Conf. (Miller and Priddy, eds.), AMS Contemp. Math Series 19 (1983), 23-37.