We model topologically complex solar magnetic fields in the form of magnetic braids and study the evolution of their topology quantified in terms of knot polynomials. As it is not possible to define a single knot polynomial for a complex and volume filling magnetic field, even in the shape of a simple braid, we compute the polynomials for an ensemble of magnetic streamlines, which gives us a distribution. As the magnetic field relaxes and simplifies over time, we observe a distinct change in the distribution from a high probability of complex polynomials to a larger probability of simpler ones.

1. Polynomial knot invariants
in the dynamics of
braided magnetic fields
Simon Candelaresi, David MacTaggart

2. Solar Magnetic Field
(Trace) (Trace)
2
Twisted flux tubes may rise to the
corona. (Prior and MacTaggart 2016).

3. Coronal Magnetic Fields
NASA
(Thiffeault et al. 2006)
Field line tangling in solar magnetic fields.
3

4. Magnetic Helicity
Conservation of magnetic helicity:
magnetic resistivity
Realizability condition:
Magnetic energy is bound from
below by magnetic helicity.
4
link twist knot braid

5. Beyond Magnetic Helicity
5
(Dabrowski-Tumanski et al. (2020))
Describe knots, braids and links using knot polynomials:
Jones polynomials for the trefoil knot:
closure
Use Python package Topoly to
find polynomials.
(Yeates 2011)

6. Knots and Links as Braids
6
Periodic boundaries.

7. Knots and Links as Braids
4-foil knot
need braid representation of knots and links
7

8. MHD Simulations
8
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initial condition: braids
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isothermal compressible gas
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viscous medium
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periodic in z
Pencil Code

9. Link Spectrum
9
Pick a few random field lines and determine the link type.
Time dependent spectrum of links.
Repeat ca. 200,000 times for each snapshot.

10. Trefoil Knot
10
generally simple
generally complex

11. Borromean Rings
11

12. Helical 3 Rings
12

13. Non-Helical 3 Rings
13

14. Conclusions
simon.candelaresi@gmail.com
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Knot polynomials for braids (coronal magnetic loops).
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Reconnection leads to simplification.
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Simple knots/links preserve more easily.
simon.candelaresi@gmail.com
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Single number from link distribution?
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Solar atmosphere: potential field closure?
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Sudden changes in spectrum related to violent solar events?