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Locality and the uniqueness of quantum invariants

Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA

E-mail Address: charles-frohman@uiowa.edu

https://doi.org/10.1142/S0218216519500792Cited by:0 (Source: Crossref)

Abstract

We introduce the notion of a “state function” for framed tangles in a disk. After choosing a finite set of states for each marked disk, a state function is a projection from the vector space spanned by all tangles to the vector space spanned by the states, that is local, and topologically invariant. Given the states for the Kauffman bracket, and the quantum $S U (3)$ -invariant we classify all state functions, and then compare our results to the literature.

Keywords:

AMSC: 57M27

References

1. S. Cautis, J. Kamnitzer and S. Morrison, Webs and quantum skew Howe duality, Math. Ann. 360(1–2) (2014) 351–390. Crossref, Web of Science, Google Scholar
2. R. H. Crowell and R. H. Fox, Introduction to Knot Theory, Graduate Texts in Mathematics, Vol. 57 (Springer-Verlag, New York, 1977), x+182 pp., Reprint of the 1963 original. Google Scholar
3. K. Hoffman and R. Kunze, Linear Algebra, 2nd edn. (Prentice-Hall, Inc., Englewood Cliffs, N.J. 1971), viii+407 pp. Google Scholar
4. V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126(2) (1987) 335–388. Crossref, Web of Science, Google Scholar
5. L. H. Kauffman, State models and the Jones polynomial, Topology 26(3) (1987) 395–407. Crossref, Web of Science, Google Scholar
6. L. H. Kauffman, On Knots, Annals of Mathematics Studies, Vol. 115 (Princeton University Press, Princeton, NJ, 1987), xvi+481 pp. Google Scholar
7. L. H. Kauffman, Knots and Physics, 4th edn. Series on Knots and Everything (World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013), xviii+846 pp. Link, Google Scholar
8. G. Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180(1) (1996) 109–151. Crossref, Web of Science, Google Scholar
9. M. Khovanov, $s l (3)$ link homology, Algebr. Geom. Topol. 4 (2004) 1045–1081. Crossref, Google Scholar
10. W. B. R. Lickorish, What is a skein module?. Google Scholar
11. W. B. R. Lickorish, An Introduction to Knot Theory, Graduate Texts in Mathematics, Vol. 175 (Springer-Verlag, New York, 1997), x+201 pp. Crossref, Google Scholar
12. S. Morrison, E. Peters and N. Snyder, Categories generated by a trivalent vertex, Selecta Math. (N.S.) 23(2) (2017) 817–868. Crossref, Web of Science, Google Scholar
13. H. Murakami, T. Ohtsuki and S. Yamada, Homfly polynomial via an invariant of colored plane graphs, Enseign. Math. (2) 44(3–4) (1998) 325–360. Google Scholar
14. T. Ohtsuki and S. Yamada, Quantum SU(3) invariant of 3-manifolds via linear skein theory, J. Knot Theory Ramifications 6(3) (1997) 373–404. Link, Web of Science, Google Scholar
15. N. Yu. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127(1) (1990) 1–26. Crossref, Web of Science, Google Scholar
16. A. S. Sikora, Skein theory for $SU (n)$ -quantum invariants, Algebr. Geom. Topol. 5 (2005) 865–897. Crossref, Web of Science, Google Scholar
17. A. S. Sikora and B. W. Westbury, Confluence theory for graphs, Algebr. Geom. Topol. 7 (2007) 439–478. Crossref, Web of Science, Google Scholar
18. E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121(3) (1989) 351–399. Crossref, Web of Science, Google Scholar