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Knot theory realizations in nematic colloids.

Simon Čopar1, Uroš Tkalec2, Igor Muševič3

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Nematic braids, formed by disclination loops in liquid crystals, are studied using colloidal particles. Their topology and knot invariants are linked to observable physical features and liquid crystal physics.

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Area of Science:

  • Soft Matter Physics
  • Liquid Crystal Science
  • Knot Theory

Background:

  • Nematic braids are complex topological structures formed by disclination loops entangling colloidal particles in liquid crystals.
  • These structures exist in thin twisted nematic layers stabilized by colloidal particle arrays.
  • Controlling these colloidal particles with laser tweezers allows for experimental manipulation of nematic braids.

Purpose of the Study:

  • To investigate entangled nematic disclinations in controlled colloidal systems.
  • To demonstrate the correspondence between knot invariants and observable physical features of nematic braids.
  • To explore how the nematic medium's properties influence knot topology and phase diagrams.

Main Methods:

  • Experimental assembly of nematic braid structures using colloidal particles in a liquid crystal.
  • Application of laser tweezers for precise control of colloidal particle arrangements.
  • Analysis of disclination loop topology using knot invariants, constructed graphs, and associated surfaces.

Main Results:

  • Established a clear correspondence between knot invariants and physically observable geometric features of the nematic braids.
  • Identified additional topological parameters introduced by the nematic medium that couple with knot topology.
  • Demonstrated that crystalline order simplifies the construction of the Jones polynomial and medial graphs.

Conclusions:

  • The study links the abstract concepts of knot theory to the physical reality of nematic liquid crystals.
  • The nematic environment introduces unique topological parameters that influence the phase diagram of possible braid structures.
  • The physics of liquid crystals mirrors the algorithmic steps in constructing knot theory tools like the Jones polynomial.