The Sigma 10-20mm 1:4-5.6 EX DC HSM is a design introduced in 2005.

Your Nikon D7100 was introduced in 2013.

Thus, Sigma had no way to test the design of that lens to see if it would work with the D7100.

This is all based upon the way that third party lens makers design lenses for other manufacturers' cameras. They do not usually license technology from the camera makers. Instead, they "reverse engineer" the camera maker's protocols by obtaining examples of cameras and lenses and measuring how they work. They measure the signals moving between the camera and lens. They may try to extract code from the camera or lens' data storage components. Eventually they try to produce a lens that can "understand" what the camera sends to the lens and can "mimic" what the camera maker's lenses send back to the camera.

They then test their prototype lens on each of the camera models available from the camera maker at that time. They have no way of testing their new lens with camera models that will be introduced in the future.

The main issue that sometimes comes up with this approach is that the camera makers may not utilize all of the code contained in their protocol with every camera and lens model currently on the market. They may have certain pieces of code included that are reserved for future use with cameras and/or lenses currently under development. They may have code intended for projects they hope to do even further into the future that may or may not ever come to fruition.

If the camera maker starts utilizing some of this "hidden" code in products produced after the third party lens was "reverse engineered" it can often have detrimental effects on the functionality of the older third party lens used with a newer camera body. Sometimes lens makers have subsequently updated the firmware for an older lens to make it compatible with newer camera bodies. This required the lens to be sent to a service center for the lens' memory chips to be swapped out or reflashed (overwritten with the new code). Newer copies of the lens made after the newer camera models were introduced may or may not have had the newer firmware when shipped, and of course older copies of the lens that have been sitting in a retailer's inventory for several years would not have the update if sold after the newer firmware had been released.

Lately, though, some lens makers have started to make it possible for the end users of third party lenses to update their lenses themselves via a "dock" that attaches to the lens camera mount and connects to a computer. These docks only work with newer lenses that are designed to allow them to communicate with the dock. Tamron calls their dock and software system the "TAP-in Console" while Sigma calls their dock the more mundane "USB Dock" and the software is named "Sigma Optimization Pro."

Unfortunately for owners of older lenses such as the Sigma 10-20mm 1:4-5.6 EX DC HSM, the new docks only work with lenses that are specifically made to be compatible with these USB docks and were introduced at around the same time or after the introduction of the docks. Sigma introduced their USB dock system in May of 2013 and it is compatible with their 'Global Vision' series of lenses that include models in the 'Art', 'Sports', and 'Contemporary' lines.

At one time Sigma offered a firmware update to the early examples of your lens (later copies made after this issue was known and resolved got the updated firmware when they were made) to make it compatible with the D5300 and similar newer Nikon cameras. They did this at no cost for lenses under warranty and for a minimal fee (about £40 in Belgium in 2016). The owner was responsible for shipping the lens to a Sigma Service Center. You should contact the Sigma Service center for your geographic location and inquire if that update is still available.