6. Discussion and conclusion

According to the diffeomorphism invariance of winding space-time, there exists symmetry between the entanglement transformation along different directions. Therefore, with the help of a complete set of definition of reduced Gamma matrices γμ, there exist other topological

acterizes the deformation of a knot (an SO(4)/SO(3) topological defect) on a 3 + 1D zero-lattice, the upper index of the topological mutual BF term Rij ∧ e<sup>k</sup> ∧ e<sup>l</sup> must be symmetric, i.e.,

By considering the SO(3,1) Lorentz invariance, the topological mutual BF term SMBF turns into

16πð Þa 2 ð

where <sup>G</sup> is the induced Newton constant which is <sup>G</sup> <sup>¼</sup> <sup>a</sup>2. The role of the Planck length is

Finally, from above discussion, we derived an effective theory of knots on deformed zero-

x

As a result, in continuum limit a knot-crystal becomes a space-time background like smooth manifold with emergent Lorentz invariance, of which the effective gravity theory turns into

For emergent gravity in knot physics, an important property is topological interplay between zero-lattice and knots. From the Einstein-Hilbert action, we found that the key property is duality between Riemann curvature Rij and strength of auxiliary gauge field Fkl: the deformation

In addition, there exist a natural energy cutoff ℏω<sup>0</sup> and a natural length cutoff a. In high energy limit (Δω � ω0) or in short range limit (Δx � a), without well-defined 3 + 1D zero-lattice, there

Ψ e μ <sup>a</sup> γ<sup>a</sup>

<sup>ð</sup> ffiffiffiffiffiffi �<sup>g</sup> <sup>p</sup> Rd<sup>4</sup>

<sup>R</sup>μν � <sup>1</sup> 2

<sup>ð</sup> ffiffiffiffiffiffi �<sup>g</sup> <sup>p</sup> Rd<sup>4</sup> eijklRij ∧ e

x

Db <sup>μ</sup> � mknot � �

b∂<sup>μ</sup> þ iωμ. The variation of the action S via the metric δgμν gives the Einstein's

<sup>k</sup> ∧ e l

Ψd<sup>4</sup> x:

Rgμν ¼ 8πGTμν: (133)

<sup>i</sup> SMBF,i that char-

(131)

(132)

mutual BF terms <sup>S</sup>MBF,i. For the total topological mutual BF term <sup>S</sup>MBF <sup>¼</sup> <sup>P</sup>

<sup>S</sup>MBF <sup>¼</sup> <sup>S</sup>EH <sup>¼</sup> <sup>1</sup>

<sup>¼</sup> <sup>1</sup> 16πG

played by lp ¼ a, that is the "lattice" constant on the 3 + 1D zero-lattice.

S ¼ Szero�lattice þ SEH

ð ffiffiffiffiffiffiffiffiffiffiffiffi �g xð Þ q

¼

þ 1 16πG

of entanglement pattern leads to the deformation of space-time.

does not exist emergent gravity at all.

i, j, k, l ¼ 1; 2; 3; 0.

62 Superfluids and Superconductors

the Einstein-Hilbert action SEH as

lattice in continuum limit as

where Db <sup>μ</sup> ¼ i

topological field theory.

equations

In this paper, we pointed out that owing to the existence of local Lorentz invariance and diffeomorphism invariance there exists emergent gravity for knots on 3 + 1D zero-lattice. In knot physics, the emergent gravity theory is really a topological theory of entanglement deformation. For emergent gravity theory in knot physics, a topological interplay between 3 + 1D zero-lattice and the knots appears: on the one hand, the deformation of the 3 + 1D zerolattice leads to the changes of knot-motions that can be denoted by curved space-time; on the other hand, the knots trapping topological defects deform the 3 + 1D zero-lattice that indicates matter may curve space-time. The Einstein-Hilbert action SEH becomes a topological mutual BF term SMBF that exactly reproduces the low energy physics of the general relativity. In Table 1, we emphasize the relationship between modern physics and knot physics.

In addition, this work would help researchers to understand the mystery in gravity. In modern physics, matter and space-time are two different fundamental objects and matter may move in (flat or curved) space-time. In knot physics, the most important physics idea for gravity is the unification of matter and space-time, i.e.,

$$\text{Matter } (\text{knots}) \Leftrightarrow \text{Space-time} (\text{zero-lattice}).\tag{134}$$

One can see that matter (knots) and space-time (zero-lattice) together with motion of matter are unified into a simple phenomenon—entangled vortex-membranes and matter (knots) curves space-time (3 + 1D zero-lattice) via a topological way.

In the end of the paper, we address the possible physical realization of a 1D knot-crystal based on quantized vortex-lines in <sup>4</sup> He superfluid. Because the emergent gravity in knot physics is topological interplay between zero-lattice and knots, there is no Einstein gravity on a 1D knotcrystal based on entangled vortex-lines in <sup>4</sup> He superfluid. However, the curved space-time could be simulated.

Firstly, we consider two straight vortex-lines in <sup>4</sup> He superfluid between opposite points on the system. Then, we rotate one vortex line around another by a rotating velocity ω0. Now, the


Table 1. The relationship between modern physics and knot physics.

winding vortex-line becomes a helical one described by r0eik0�x�iω0tþiϕ<sup>0</sup> with ω<sup>0</sup> ≃ <sup>κ</sup> <sup>4</sup><sup>π</sup> ln <sup>1</sup> k0a<sup>0</sup> <sup>k</sup> 2 0. As a result, a knot-crystal is realized. For <sup>4</sup> He superfluid, κ ¼ h=m is the discreteness of the circulation owing to its quantum nature [2]. h is Planck constant and m is atom mass of SF. So <sup>κ</sup> <sup>¼</sup> <sup>h</sup>=<sup>m</sup> is about 10�<sup>3</sup> cm<sup>2</sup> /s. The length of the half pitch of the windings <sup>a</sup> <sup>¼</sup> <sup>π</sup> <sup>k</sup><sup>0</sup> is set to be 10�<sup>5</sup> cm, and the distance between two vortex lines r<sup>0</sup> is set to be 10�<sup>6</sup> cm. We then estimate the effective light speed <sup>c</sup>eff that is defined by <sup>c</sup>eff <sup>¼</sup> <sup>κ</sup>k<sup>0</sup> <sup>2</sup><sup>π</sup> ln <sup>1</sup> <sup>k</sup>0a<sup>0</sup> � <sup>1</sup> 2 (a<sup>0</sup> denotes the vortex filament radius which is much smaller than any other characteristic size in the system). The effective light speed ceff is about 4 m/s. A non-uniform winding length leads to an effective curved 1 + 1D space-time.

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Topological Interplay between Knots and Entangled Vortex-Membranes

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However, at finite temperature, there exist mutual friction and phonon radiation for Kelvin waves on quantized vortex-lines in <sup>4</sup> He superfluid. After considering these dissipation effects, the Kelvin waves are subject to Kolmogorov-like turbulence (even in quantum fluid [3, 4]).
