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Abstract The entanglement entropy of many quantum systems is difficult to compute in general. They are obtained as a limiting case of the Rényi entropy of index m, which captures the higher moments of the reduced density matrix. In this work, we study pure bipartite states associated with S 3 complements of a two-component link which is a connected sum of a knot K $$ \mathcal{K} $$ and the Hopf link. For this class of links, the Chern-Simons theory provides the necessary setting to visualise the m-moment of the reduced density matrix as a three-manifold invariant Z( M K m $$ {M}_{{\mathcal{K}}_m} $$ ), which is the partition function of M K m $$ {M}_{{\mathcal{K}}_m} $$ . Here M K m $$ {M}_{{\mathcal{K}}_m} $$ is a closed 3-manifold associated with the knot K $$ \mathcal{K} $$ m , where K $$ \mathcal{K} $$ m is a connected sum of m-copies of K $$ \mathcal{K} $$ (i.e., K $$ \mathcal{K} $$ # K $$ \mathcal{K} $$ . . . # K $$ \mathcal{K} $$ ) which mimics the well-known replica method. We analayse the partition functions Z( M K m $$ {M}_{{\mathcal{K}}_m} $$ ) for SU(2) and SO(3) gauge groups, in the limit of the large Chern-Simons coupling k. For SU(2) group, we show that Z( M K m $$ {M}_{{\mathcal{K}}_m} $$ ) can grow at most polynomially in k. On the contrary, we conjecture that Z( M K m $$ {M}_{{\mathcal{K}}_m} $$ ) for SO(3) group shows an exponential growth in k, where the leading term of ln Z( M K m $$ {M}_{{\mathcal{K}}_m} $$ ) is the hyperbolic volume of the knot complement S 3 \ K $$ \mathcal{K} $$ m . We further propose that the Rényi entropies associated with SO(3) group converge to a finite value in the large k limit. We present some examples to validate our conjecture and proposal.

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 W ̂ 3 $$ {\hat{W}}_3 $$ (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving ℛ $$ \mathrm{\mathcal{R}} $$ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional ℛ ̂ $$ \hat{\mathrm{\mathcal{R}}} $$ -matrices can be written in terms of infinite family of Laurent polynomials V n , t q $$ {\mathcal{V}}_{n,t}\left[q\right] $$ whose absolute coefficients has interesting relation to the Fibonacci numbers ℱ n $$ {\mathrm{\mathcal{F}}}_n $$ . 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.

Abstract We study the multi-boundary entanglement structure of the state associated with the torus link complement S 3 \T p,q in the set-up of three-dimensional SU(2) k Chern-Simons theory. The focal point of this work is the asymptotic behavior of the Rényi entropies, including the entanglement entropy, in the semiclassical limit of k → ∞. We present a detailed analysis of several torus links and observe that the entropies converge to a finite value in the semiclassical limit. We further propose that the large k limiting value of the Rényi entropy of torus links of type T p,pn is the sum of two parts: (i) the universal part which is independent of n, and (ii) the non-universal or the linking part which explicitly depends on the linking number n. Using the analytic techniques, we show that the universal part comprises of Riemann zeta functions and can be written in terms of the partition functions of two-dimensional topological Yang-Mills theory. More precisely, it is equal to the Rényi entropy of certain states prepared in topological 2d Yang-Mills theory with SU(2) gauge group. Further, the universal parts appearing in the large k limits of the entanglement entropy and the minimum Rényi entropy for torus links T p,pn can be interpreted in terms of the volume of the moduli space of flat connections on certain Riemann surfaces. We also analyze the Rényi entropies of T p,pn link in the double scaling limit of k → ∞ and n → ∞ and propose that the entropies converge in the double limit as well.

Abstract Topological entanglement structure amongst disjoint torus boundaries of three manifolds have already been studied within the context of Chern-Simons theory. In this work, we study the topological entanglement due to interaction between the quasiparticles inside three-manifolds with one or more disjoint S 2 boundaries in SU(N) Chern-Simons theory. We focus on the world-lines of quasiparticles (Wilson lines), carrying SU(N) representations, creating four punctures on every S 2. We compute the entanglement entropy by partial tracing some of the boundaries. In fact, the entanglement entropy depends on the SU(N) representations on these four-punctured S 2 boundaries. Further, we observe interesting features on the GHZ-like and W-like entanglement structures. Such a distinction crucially depends on the multiplicity of the irreducible representations in the tensor product of SU(N) representations.

Abstract We study the entanglement for a state on linked torus boundaries in 3d Chern-Simons theory with a generic gauge group and present the asymptotic bounds of Rényi entropy at two different limits: (i) large Chern-Simons coupling k, and (ii) large rank r of the gauge group. These results show that the Rényi entropies cannot diverge faster than ln k and ln r, respectively. We focus on torus links T (2, 2n) with topological linking number n. The Rényi entropy for these links shows a periodic structure in n and vanishes whenever n = 0 (mod p), where the integer p is a function of coupling k and rank r. We highlight that the refined Chern-Simons link invariants can remove such a periodic structure in n.

Abstract Tests of the integrality properties of a scalar operator in topological strings on a resolved conifold background or orientifold of conifold backgrounds have been performed for arborescent knots and some non-arborescent knots. The recent results on polynomials for those knots colored by SU(N ) and SO(N ) adjoint representations [1] are useful to verify Marino’s integrality conjecture up to two boxes in the Young diagram. In this paper, we review the salient aspects of the integrality properties and tabulate explicitly for an arborescent knot and a link. In our knotebook website, we have put these results for over 100 prime knots available in Rolfsen table and some links. The first application of the obtained results, an observation of the Gaussian distribution of the LMOV invariants is also reported.