We develop a Mellin transform framework which allows us to simultaneously analyze the four known exactly solvable 1`1 dimensional lattice polymer models: the log-gamma, strict-weak, beta, and inversebeta models. Using this framework we prove the conjectured fluctuation exponents of the free energy and the polymer path for the stationary point-to-point versions of these four models. The fluctuation exponent for the polymer path was previously unproved for the strict-weak, beta, and inverse-beta models.

Motivated by the study of directed polymer models with random weights on the square integer lattice, we define an integrability property shared by the log-gamma, strict-weak, beta, and inverse-beta models. This integrability property encapsulates a preservation in distribution of ratios of partition functions which in turn implies the so called Burke property. We show that under some regularity assumptions, up to trivial modifications, there exist no other models possessing this property. 1 this property is the log-gamma model, introduced by Seppäläinen in [14]. In his paper T h,Y -invariance is used to prove the conjectured values for the fluctuation exponents of the free energy and the polymer path in the stationary point-to-point case and to prove upper bounds for the exponents in the point-to-point and point-to-line cases without boundary conditions. In [8] Georgiou and Seppäläinen use T h,Y -invariance to obtain large deviation results for the log-gamma polymer. In the setting of directed polymer models, this is the first instance where precise large deviation rate functions for the free energy were derived.Thereafter three additional models admitting T h,Y -invariant versions were found: the strict-weak model, introduced simultaneously by Corwin, Seppäläinen, and Shen in [7] and O'Connell and Ortmann in [12], the beta model, introduced by Barraquand and Corwin in [3] as the beta RWRE, and the inverse-beta model, introduced by Thiery and Le Doussal in [18]. The stationary versions of these models were given by Balázs, Rassoul-Agha, and Seppäläinen in [2] for the beta model, Thiery in [17] for the inverse-beta model, and by Corwin, Seppäläinen, and Shen in [7] for the strict-weak model.In this paper we present a uniqueness result for T h,Y -invariant models. That is, under some regularity assumptions and up to the two natural modifications of reflection and scaling, the log-gamma, strict-weak, beta, and inverse-beta are the only T h,Y -invariant models.In the forthcoming paper [5] we use T h,Y -invariance along with a Mellin transform framework to simultaneously prove the conjectured value for the fluctuation exponent of the free energy and the upper bound for the polymer path fluctuations in the stationary point-to-point version of these four models.

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In the simple hexagonal lattice, bridge number is used to establish a lower bound on stick numbers of knots. This result aids in giving a new proof that the minimal stick number is 11. In addition, the authors establish upper bounds for the stick number of a composite knot. Constructions for (p, p+1)-torus knots and some 3-bridge knots are given requiring one more stick than the lower bound guarantees.

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