adverbs - What's the meaning of the く in 赤くありません?

The way of turning 赤 into an adjective is adding い, but what does it mean when instead of a い you put a く？

I found it in this phrase:

私のリンゴは赤くありません。

I get the meaning but not the meaning of く.

Answer

Keep in mind that Japanese "adjectives" function differently from English adjectives. You are in this case asking us about a い-adjective, 赤い. Those adjectives differ from the な-adjectives, for example 有名｛ゆうめい｝な:

• The い or な is attached to those adjectives when used as epithets, i.e. they are attached to a noun:
  

  
  

  有名な人、赤いリンゴ.

  
  

  
  


• な-adjectives are used with the copula, dropping the な:
  
  

  
  

  この人は有名(です・でした・ではありません)

  
  

  
  


• 
  

  い-adjectives are "conjugated", you can use them without modifying the copula (sometimes even remove it entirely)! In your example

  
  

  
  

  このリンゴは赤い(です)。

  
  

  
  

  If you to turn this sentence into a negative one, you modify the adjective, by appending ～くない, i.e. このリンゴは赤くない(です). See for example this page or this one.

  
  
  

  
  

  このリンゴは赤いではありません。

  
  

  
  

  is not a correct sentence.

  
  

Then, as mentioned in the related answer ～くありません is the polite form of ～くない. Thus, it is the polite negative form of the adjective.