What happens if a metal wire that was one-atom in thickness was pulled across a finger? What would happen if a metal wire that was one-atom was pulled across your finger? Would it cut off your finger, or would it pass through your finger without harming you? What if the metal one atom thick was "unbreakable"?
 A: A one atom thick wire will not cause any damage to the finger whatsoever.
A free hanging one-atom thick wire won’t even be stable.
Regardless, let’s examine what happens when you apply tiny force on your finger. Let’s take a special case: you have a scab on your finger. You’re trying to cut your finger with a scab with an “unbreakable” one-atom thick wire; let’s see what happens.
Below is a molecule of fibrin, the molecule responsible for blood clots (scabs). Fibrinogen is a molecule that circulates in blood. When there is an injury. Fibrinogen gets converted to a network of fibers called fibrin. This is the chief ingredient of a scab.

Image Description

((A) Model of human fibrinogen based on the crystal structure modified from [19]. (B) Model of half-staggered, double-stranded
  fibrin protofibril, illustrating the knob-hole interactions ‘A:a’.
  Because the locations of the αC regions in the protofibril are
  unknown, they are omitted from this model. (E) The model in panel C
  superimposed on a transmission electron micrograph of a
  negatively-stained fibrin fiber segment with alignment of the stain
  pattern with the assembled fiber pattern.)

So what happens when you apply a tiny force on fibrin? Well there are many studies and a few of the involved applying a tiny pressure of under a nano-Newton to the fibrin molecule using Atomic Force Microscope tip that’s only a few atoms sharp. So what happened to fibrin molecule when one did that? It unfolded. Fibrin, is a protein, and like all proteins, it unfolded under certain conditions. The condition here being applying mechanical pressure. And the resulting unfolded protein has actually lower mechanical strength that the non unfolded protein.

Image Description

((A) Stress-strain curves of cylindrical fibrin clots. Data are shown with solid black like, fitted model in red. 1 (B) Force-induced
  unfolding of the γ-nodule measured by AFM. Distances between knob ‘A’
  and hole ‘a’ are plotted relative to event 2. The clusters of forces
  representing the four events in the complex force pattern are labeled
  with numbers.)

By needling the fibrin was lower its mechanical strength
But our little needle (the typical AFM tip) doesn’t qualify as a “one atom thick” wire.

Image Description

Field emission SEM image of Single Walled Nano Tube bundles grown
  from a Si cantilever tip assembly

This is how close we could get to a “one-atom thick” wire. It’s several nano-meters thick. And trust me these things are delicate. You’re own breath will destroy the wire. People handling such intricate materials must wear a lot of protection (protection for the materials!). There is no need to look up mechanical properties of Nano-wires because none will survive nano-indentation experiments. These things are so delicate that they don’t even induce protein folding by mechanical pressure (by inducing folding of fibrin because they self destruct at those pressures).
And finally, what about a “one atom thick” wire? You can’t make one! You see, metals need to bond with atoms in all directions and hence won’t be stable in a free hanging one atom thick wire. So there can’t be a “one atom thick” wire, let alone something unbreakable.
So, summing up:


*

*You can’t make a one-atom thick wire, let alone something “unbreakable”.

*Wires that are hundreds of atoms thick won’t cause any damage. They self-destruct on contact.

*Tiny forces applied to fibrin by structures a few thousand atoms thick (AFM tips) only weaken the scabs (fibrin network).
References
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2975759/
http://www.pnas.org/content/97/8/3809.full.pdf 
