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Observations suggest that twisted magnetic flux tubes are ubiquitous in theSun's atmosphere. The main aim of this work is to advance the study ofaxisymmetric modes of magnetic flux tubes by modeling both twisted internal andexternal magnetic field, when the magnetic twist is weak. In this work, wesolve the derived wave equations numerically assuming twist outside the tube isinversely proportional to the distance from its boundary. We also study thecase of constant magnetic twist outside the tube and solve these equationsanalytically. We show that the solution for a constant twist outside the tubeis a good approximation to the case where the magnetic twist is proportional to$1/r$, namely the error is in all cases less than $5.4\%$.The solution is inexcellent agreement with solutions to simpler models of twisted magnetic fluxtubes, i.e. without external magnetic twist. It is shown that axisymmetricAlfv\'{e}n waves are naturally coupled with magnetic twist as the azimuthalcomponent of the velocity perturbation is nonzero. We compared our theoreticalresults with observations and comment on what the Doppler signature of thesemodes is expected to be. Lastly, we argue that the character of axisymmetricwaves in twisted magnetic flux tubes can lead to false positives in identifyingobservations with axisymmetric Alfv\'{e}n waves.

Axisymmetric Modes in Magnetic Flux Tubes with Internal and External Magnetic Twist