Tangle Equations, the Jones conjecture, slopes of surfaces in tangle complements, and q-deformed rationals

DOI10.4153/S0008414X23000755arXiv2005.08162OpenAlexW4388661112MaRDI QIDQ6340856

Publication date: 17 May 2020

Abstract: We study systems of

2

-tangle equations

N(X+T_1)=L_1,quad N(X+T_2)=L_2,

which play an important role in the analysis of enzyme actions on DNA strands. We show the benefits of considering such systems in the context of framed tangles and, in particular, we conjecture that in this setting each such system has at most one solution

X .

We prove a version of this statement for rational tangles. More importantly, %we establish a connection between systems of tangle equations and the Jones conjecture. Specifically, we show that the Jones conjecture implies that if a system of tangle equations has a rational solution then that solution is unique among all tangles. This result potentially opens a door to a purely topological line of attack on the Jones conjecture. Additionally, we relate systems of tangle equations to the Cosmetic Surgery Conjecture. Furthermore, we establish a number of properties of the Kauffman bracket

([T]_{0}, [T]_{i} n f t y)

of

2

-tangles

T

for the purpose of the proofs of the above results, which are of their own independent interest. In particular, we show that the Kauffman bracket ratio

Q (T) = [T]_{i} n f t y / [T]_{0}

quantizes continued fraction expansions of rationals. Furthermore, we prove that for algebraic tangles

Q (T)

determines the slope of incompressible surfaces in

D^{3} s m a l l s e t m i n u s T

.

Full work available at URL: https://doi.org/10.4153/s0008414x23000755

Mathematics Subject Classification ID

Knot polynomials (57K14) Knot theory (57K10) General topology of 3-manifolds (57K30)