In this work, we explore the combinatorics arising from the quiver generating series of the unreduced r-colored HOMFLY-PT polynomial P̅_r(a,q) for some twist-knots and double twist knots. By taking the limit a = 0 and q = 1, we indeed obtain lattice path models for these knots.

A novel duality sequence is devised to study late‐time cosmology in the heterotic setup of Horava and Witten with dynamical walls that are moving towards each other. Remarkably, the dimensionally reduced 4‐dimensional theory does not violate NEC and no bouncing or ekpyrotic phase is observed. Instead, the 4‐dimensional setup shows a transient de Sitter phase that lies well within the trans‐Planckian bound. This opens up a myriad of possibilities of addressing both phenomenological and cosmological issues, with focus on an axionic cosmology with temporally varying axionic coupling.

SL(2,ℂ) Chern-Simons theory on a closed 3-manifold is one of the most interesting, yet tractable examples of a QFT. On one hand, its non-perturbative structure is not yet fully understood; on the other, the mathematical structure turns out to be very rich. In this work we explore the new phenomenon of flat connections at infinity on various knot surgery manifolds. Such flat connections can be understood as asymptotic ends in the non-compact moduli space of flat SL(2,ℂ) connections. We focus on the examples of ± 1/r-surgeries on torus, twist and some double twist knot complements in S^3. Surprisingly, our findings suggest that flat connections at infinity are abundant even for simple low-crossing knot surgeries. We therefore believe that their presence would shed light on the resurgent nature of the path integral.

One of the long-standing puzzles in string theory has been on the existence of a four-dimensional de Sitter and quasi de Sitter configurations, the latter being defined with a temporally varying dark energy, in E_8 x E_8 and SO(32) heterotic theories. In this work, novel dynamical duality-sequences are devised that provide natural constructions of de Sitter and quasi de Sitter excited states in the aforementioned theories allowing no late-time singularities. The emergent positive dark energies -- including the intriguing possibility of their slow temporal variations -- appear from Borel resumming Gevrey series from the path-integral representations of such states. Additionally, precise ways to handle the equations of motion, Bianchi identities, flux quantizations and anomaly cancellations -- consistent with the underlying axionic cosmology and with the probability of forming wormholes that connect baby universes -- are presented for the SO(32) and the E_8 x E_8 theories within a framework that systematically incorporates perturbative and non-perturbative corrections in the far infrared. The temporally varying dark energy, which is much more natural in our set-up because of its emergent nature, surprisingly simplifies many of the aforementioned computations. Interestingly, our analysis also provides, probably for the first time, a set-up to consistently embed four-dimensional standard model degrees of freedom at late time in a realistic gravitational background with positive dark energy from string theory.

We relate the stability of knot invariants under twisting a pair of strands to the stability of symmetric quivers under unlinking (or linking) operation. Starting from the HOMFLY-PT skein relations, we confirm the stable growth of Sym^r-coloured HOMFLY-PT polynomials under the addition of a full twist to the knot. On the other hand, we show that symmetric quivers exhibit analogous stable growth under unlinking or linking of the quiver augmented with the extra node; in some cases this augmented quiver captures the spectrum of motivic Donaldson-Thomas invariants of all quivers in the sequence. Combining these two versions of the stable growth, we conjecture that performing a full twist on any knot corresponds to appropriate unlinking or linking of the corresponding augmented quiver – this statement is an important step towards a direct definition of the knot-quiver correspondence based on the knot diagram. We confirm the conjecture for all twist knots, (2,2p+1) torus knots, and all pretzel knots up to 15 crossings with an odd number of twists in each twist region.

We propose a modification in the Bethe-like ansatz to reproduce the hydrogen atom spectrum and the wave functions. Such a proposal provided a clue to attempt the exact quantization condition (EQC) for the quantum periods associated with potentials V(x)V(x) which are of the form (V(x) = |x| + a/|x| + b/|x|^2 )(V(x)=|x|+a/|x|+b/|x|2). We validate the EQC proposal by showing that our computed Voros spectrum in the limit a,b → 0a,b→0 is matching well with the true spectrum of the familiar |x||x| potential. Thus we have given a route to obtain the spectral solution for the one dimensional Schrödinger equation involving potentials with regular singularity at the origin.

We devise a novel duality sequence to study late-time cosmology in the heterotic E_8 x E_8 setup of Horava and Witten with dynamical walls that are moving towards each other. Surprisingly, we find that the dimensionally reduced four-dimensional theory does not violate NEC and therefore we do not see either a bouncing or an ekpyrotic phase. Instead, our four-dimensional setup shows a transient de Sitter phase that lies well within the trans-Planckian bound. This opens up a myriad of possibilities of addressing both phenomenological and cosmological issues, and here we concentrate on one such interesting model, an axionic cosmology with temporally varying axionic coupling.

We propose an algebraic procedure to obtain U q ( sl 3 ) quantum 3 j -symbols (quantum Clebsch–Gordan coefficients) appearing in the decomposition of tensor product of symmetric representations. In fact, these symbols will be useful to write the spectral parameter dependent R -matrix elements for any bi-partite vertex model whose edges carry states of the symmetric representations.

We compute the PSL(2,ℂ) Chern-Simons partition function of a closed 3-manifold obtained from Dehn fillings of the link complement 𝐒^3\ℒ, where ℒ=𝒦# H is the connected sum of the knot 𝒦 with the Hopf link H. Motivated by our earlier work on topological entanglement and the reduced density matrix σ for such link complements, we wanted to determine a choice of Dehn filling so that the trace of the matrix σ becomes equal to the PSL(2,ℂ) partition function of the closed 3-manifold. We use the SnapPy program and numerical techniques to show this equivalence up to the leading order. We have given explicit results for all hyperbolic knots 𝒦 up to six crossings.

In our earlier work, we studied the Ẑ -invariant(or homological blocks) for SO(3) gauge group and we found it to be same as Ẑ^SU(2) . This motivated us to study the Ẑ -invariant for quotient groups SU(N)/ℤ_m , where m is some divisor of N. Interestingly, we find that Ẑ -invariant is independent of m.

In this paper we study $U(N)$ colored HOMFLY-PT polynomials of torus links in the double scaling limit (polynomial variable $q\rightarrow 1$, $N\rightarrow \infty$ keeping $q^N$ fixed). We show that, in this limit, the colored HOMFLY-PT polynomial of any $(L\alpha,L\beta)$ torus link can be expressed in terms of the colored HOMFLY-PT polynomial of $(L,L)$ torus link. Using the connection between matrix models and the Chern-Simons field theoretic invariants, we show that the colored torus link invariants are uniquely expressed in terms of connected correlation functions of operators in $U(N)$ matrix model. We determine the leading and subleading contribution to some of the correlators at large $N$ from the matrix model approach and find that they match exactly with those obtained from the corresponding colored HOMFLY-PT polynomials.

Recent no-go theorems have ruled out four-dimensional classical de Sitter vacua in heterotic string theory. On the other hand, the absence of a well-defined Wilsonian effective action and other related phenomena also appear to rule out such time-dependent vacua with de Sitter isometries, even in the presence of quantum corrections. In this note, we argue that a four-dimensional de Sitter space can still exist in SO(32) heterotic string theory as a Glauber-Sudarshan state, i.e. as a coherent state, over a supersymmetric Minkowski background, albeit within a finite temporal domain. Borel resummation and resurgence play a crucial role in constructing such a state in the Hilbert space of heterotic theory governed entirely by the IR degrees of freedom.

We propose an algebraic expression for $U_q(\mathfrak{sl}_3)$ quantum $3j$ symbols (quantum Clebsch-Gordan coefficients) appearing in the decomposition of tensor product of symmetric representations. Our compact form will be useful to write the spectral parameter dependent $R$-matrix elements for any bi-partite vertex model whose edges carry states of the symmetric representations.

Three-manifold invariants Ẑ (“Z-hat”), also known as homological blocks, are q-series with integer coefficients. Explicit q-series form for Ẑ is known for SU(2) group, supergroup SU(2|1) and orthosymplectic supergroup OSp(2|2). We focus on Ẑ for SO(3) group and orthosymplectic supergroup OSp(1|2) in this paper. Particularly, the change of variable relating SU(2) link invariants to the SO(3) and OSp(1|2) link invariants plays a crucial role in explicitly writing the q-series.

We propose an algebraic expression for U_q(𝔰𝔩_3) quantum 3j symbols (quantum Clebsch-Gordan coefficients) appearing in the decomposition of tensor product of symmetric representations. Our compact form will be useful to write the spectral parameter dependent R-matrix elements for any bi-partite vertex model whose edges carry states of the symmetric representations.

We obtain a quiver representation for a family of knots called double twist knots $K(p,-m)$. Particularly, we exploit the reverse engineering of Melvin-Morton-Rozansky(MMR) formalism to deduce the pattern of the charge matrix for these quivers.

Three-manifold invariants (Z) over cap ("Z-hat"), also known as homological blocks, are q-series with integer coefficients. Explicit q-series form for (Z) over cap is known for SU(2) group, supergroup SU(2 vertical bar 1) and orthosymplectic supergroup OSp(2 vertical bar 2). We focus on (Z) over cap for SO(3) group and orthosymplectic supergroup OSp(1 vertical bar 2) in this paper. Particularly, the change of variable relating SU(2) link invariants to the SO(3) and OSp(1 vertical bar 2) link invariants plays a crucial role in explicitly writing the q-series.

Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

Knot theory is actively studied both by physicists and mathematicians as it provides a connecting centerpiece for many physical and mathematical theories. One of the challenging problems in knot theory is distinguishing mutant knots. Mutant knots are not distinguished by colored HOMFLY-PT polynomials for knots colored by either symmetric and or antisymmetric representations of SU(N). Some of the mutant knots can be distinguished by the simplest non-symmetric representation [2,1]. However there is a class of mutant knots which require more complex representations like [4,2]. In this paper we calculate polynomials and differences for the mutant knot polynomials in representations [3,1] and [4,2] and study their properties.