Quantum invariants of periodic links and periodic 3-manifolds

Chen, Qi; Le, Thang T. Q.

doi:10.4064/fm184-0-4

Cited by 17 publications

(13 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the case r > 3 is prime which we denote by p , we use some techniques of Chen and Le [6,Section 6]. In fact, the argument there together with our Corollary 3.3 show that the only possible strong p -congruence (for p 5) between P and P is for p D 5 and that the only possible strong p -congruence (for p 5) between † and † is for p D 7.…”

Section: Lemma 34mentioning

confidence: 99%

Congruence and quantum invariants of 3–manifolds

Gilmer

2007

Algebr. Geom. Topol.

View full text Add to dashboard Cite

Let f be an integer greater than one. We study three progressively finer equivalence relations on closed 3-manifolds generated by Dehn surgery with denominator f : weak f -congruence, f -congruence, and strong f -congruence. If f is odd, weak f -congruence preserves the ring structure on cohomology with ‫ޚ‬ f -coefficients. We show that strong f -congruence coincides with a relation previously studied by Lackenby. Lackenby showed that the quantum S U.2/ invariants are well-behaved under this congruence. We strengthen this result and extend it to the SO.3/ quantum invariants. We also obtain some corresponding results for the coarser equivalence relations, and for quantum invariants associated to more general modular categories. We compare S 3 , the Poincaré homology sphere, the Brieskorn homology sphere †.2; 3; 7/ and their mirror images up to strong f -congruence. We distinguish the weak f -congruence classes of some manifolds with the same ‫ޚ‬ f -cohomology ring structure. 57M99; 57R56

show abstract

Section: Lemma 34mentioning

confidence: 99%

Congruence and quantum invariants of 3–manifolds

Gilmer

2007

Algebr. Geom. Topol.

View full text Add to dashboard Cite

show abstract

“…To compare our result with previous results, one has to compare the size of ideal used in Theorem 4.1 with ideals in [4,6,30,25]. As we have mentioned, we have corrected the false conjecture given by Chbili [4] and Przytycki and Sikora [30].…”

Section: Discussionmentioning

confidence: 88%

“…Furthermore, if n is odd, the ideal I n is a subset of the ideal generated by p and [3] p − [3]. To compare the ideal generated by p and To compare with the ideal in [6], we use only fundamental representations of the quantum Lie algebras sl(n) which are finite but Chen and Le used all representations of the quantum sl(n) which are obviously infinite. Thus, our criterion is sharper than other previous results.…”

Section: Discussionmentioning

confidence: 99%

“…There were subsequent studies on the conjecture [6]. But, it was shown that Conjecture 1.1 is false for n ≥ 4 [30].…”

Section: Conjecture 11 ([4]mentioning

confidence: 99%

“…In this section, we concentrate on the quantum sl(n) skein modules and prove Theorem 3.4. The authors have been trying to find a complete relation of the quantum sl(n) representation theory, but we find that the size of a polygon we have to find a suitable relation is increasing as n increases: there a rectangular relation for the quantum sl(3), a hexagonal relation for the quantum sl(4) and an octagonal relation for the quantum sl (6). In particular, a complete set of relations for the quantum sl(4) representation theory is conjectured [17].…”

Section: The Quantum Sl(n) Skein Modulesmentioning

confidence: 99%

See 2 more Smart Citations

THE QUANTUM sl(n, ℂ) REPRESENTATION THEORY AND ITS APPLICATIONS

Jeong¹,

Kim²

2012

Journal of the Korean Mathematical Society

View full text Add to dashboard Cite

Abstract. In this paper, we study the quantum sl(n) representation category using the web space. Specially, we extend sl(n) web space for n ≥ 4 as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial Pn(q) specialized to a one variable polynomial can be computed by a linear expansion with respect to a presentation of the quantum representation category of sl(n). Moreover, we correct the false conjecture [30] given by Chbili, which addresses the relation between some link polynomials of a periodic link and its factor link such as Alexander polynomial (n = 0) and Jones polynomial (n = 2) and prove the corrected conjecture not only for HOMFLY polynomial but also for the colored HOMFLY polynomial specialized to a one variable polynomial.

show abstract