SignalR - connect to websocket service from javascript without SignalR library
Before you can connect to the server there is connection negotiation happening. This is when the server sends all the data needed to send and receive messages. Without connection negotiation you won't be able to connect to the server. Once you implement connection negotiation you will be probably half into implementing a SignalR client. I wrote a blog post describing SignalR protocol which should help you understand how things works underneath and why connecting to the server with your own websocket is not straightforward (or simply impossible if you don't follow the SignalR protocol).
EDIT
The ASP.NET Core version of SignalR now allows connecting to the server with bare webSocket.
I just want to add a that it is possible to connect to ASP.NET Core version of SignalR with websocket but you have to add the magic char 30 at the end of every call you make
const endChar = String.fromCharCode(30);
socket.send(`{"arguments":["arg1"],"streamIds":[],"target":"TestMethod","type":1}${endChar}`);