We built a pool in Brooklyn about 7 years ago and the patio material was Travertine set in mortar, not dry laid. It is a honed/polished finish that does not have the cracks and fissures on the standard material.

The coping was also travertine in 6' lengths 14" wide 2" thick. This too is honed.
The color is a creamy light tan.

We installed a SWG system and also an ozone system which we updated in the last 3 months. The newer ozone system will nearly eliminate the need for chlorine. This pool is 12'x30' about 14-15,0000 gallons.

We have used quartzite before and it's a nice material even in direct sun. The color variations are nice.

Bluestone will get very hot in the direct sun, Most dark colors tend to hold the heat as well.

We have used limestone extensively and prefer it in a sunny location. The color ranges are nice and compliments other colors and materials, and it does not get hot in the sun.

Tennessee crab stone is a nice looking stone but it does flake as it is soft.

Feel free to check our Houzz site for examples of our work. There are many pools with limestone patios and coping.

Go with 5cm travertine if you can and install a mastic behind it between the travertine and the deck. I recommend Peruvian travertine but you can get good travertine from the Tennessee area as well. Out here on the west coast flagstone is popular but it’s not super comfortable on the feet or when you’re sitting around the pool unless it’s tumbled. You can also just cantilever concrete which is the most cost effective way to go.

Thank you so much for all of your help!

Mystic pools- are there stone yards you can recommend in brooklyn or near Brooklyn?

I use Braen Suupply in Haskell NJ. They're about an hour (without traffic) drive west of NYC.

You can speak with Mike D'Ercole. He can help you with your needs and give you some suggestions. Mention my company name-been doing business with them for many years. 973- 835-1419. https://www.braensupply.com/

They have a beautiful display showroom of different stone choices.