Can a super strong rope cut through the Earth's surface?

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S1nG
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Hi guys i have some issues regarding gravity and here's how it goes. Assuming 2 man having supernatural powers with limitless strength and they have a rope which is also super strong and will never tear apart. Also take into consideration that the rope, being miles apart, still remains taut.
Now this 2 guys met at the top of the Earth and they held the rope, and they started walking to the bottom of Earth, will the rope some how "cuts" into the Earth? or will it remain above the ground? I have a visual on this i appreciate any replies :)
 

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I just do not understand how it has any relationship with gravity.
By the way, where do you from?
 
The string will melt, perhaps, so close to the core of the Earth.
 
Unless the rope stretches the two guys will have to walk underground as they move towards the equator and a larger diameter Earth surface...if the "rope" is sufficiently strong and so are the guys, it could cut throuh dirt,rock,magma,etc as you suggest...likely molten rock would spew forth and eruptions along the cut woud make life really exciting!