CommunityNews

CommunityNews

A Fast, Strong, Topologically Meaningful and Fun Knot Invariant

In this paper we discuss a pair of polynomial knot invariants $Θ=(Δ,θ)$ which is:

  • Theoretically and practically fast: $Θ$ can be computed in polynomial time. We can compute it in full on random knots with over 300 crossings, and its evaluation at simple rational numbers on random knots with over 600 crossings.
  • Strong: Its separation power is much greater than the hyperbolic volume, the HOMFLY-PT polynomial and Khovanov homology (taken together) on knots with up to 15 crossings (while being computable on much larger knots).
  • Topologically meaningful: It likely gives a genus bound, and there are reasons to hope that it would do more.
  • Fun: Scroll to Figures 1.1-1.4, 3.1, and 6.2.
    $Δ$ is merely the Alexander polynomial. $θ$ is almost certainly equal to an invariant that was studied extensively by Ohtsuki, continuing Rozansky, Kricker, and Garoufalidis. Yet our formulas, proofs, and programs are much simpler and enable its computation even on very large knots.

Read in full here:

Where Next?

Popular General Dev topics Top

First poster: dimitarvp
skiftOS is a simple, handmade operating system for the x86 platform, aiming for clean and pretty APIs while keeping the spirit of UNIX. s...
New
First poster: dwaynebradley
Maybe it’s just my experience, but Object-Oriented Programming seems like a default, most common paradigm of software engineering. The on...
New
New
First poster: OvermindDL1
You can now buy a 100W USB-C cable with a built-in power meter. They’re just $20 on Amazon, and they work!
New
First poster: bot
Flipper Zero is a portable multi-tool for pentesters and geeks in a toy-like body. It loves hacking digital stuff, such as radio protocol...
New
First poster: cpgo
8 reasons to ditch Chrome and switch to Firefox. Chrome may dominate, but Firefox is a known name among browsers for a reason. Whether y...
New
First poster: bot
GitHub - Rezmason/matrix: matrix (web-based green code rain, made with love). matrix (web-based green code rain, made with love) - GitHu...
New
First poster: Korbin73
Whatever happened to Elm, anyway?. I see this question pop up quite frequently in lots of different arenas - folks are curious as to wha...
New
First poster: FatimaAdamu
Two US lawyers fined for submitting fake court citations from ChatGPT. Law firm also penalised after chatbot invented six legal cases th...
New
New

Other popular topics Top

AstonJ
Or looking forward to? :nerd_face:
503 15149 280
New
dasdom
No chair. I have a standing desk. This post was split into a dedicated thread from our thread about chairs :slight_smile:
New
AstonJ
There’s a whole world of custom keycaps out there that I didn’t know existed! Check out all of our Keycaps threads here: https://forum....
New
PragmaticBookshelf
Tailwind CSS is an exciting new CSS framework that allows you to design your site by composing simple utility classes to create complex e...
New
PragmaticBookshelf
Build highly interactive applications without ever leaving Elixir, the way the experts do. Let LiveView take care of performance, scalabi...
New
PragmaticBookshelf
Use WebRTC to build web applications that stream media and data in real time directly from one user to another, all in the browser. ...
New
foxtrottwist
A few weeks ago I started using Warp a terminal written in rust. Though in it’s current state of development there are a few caveats (tab...
New
AstonJ
We’ve talked about his book briefly here but it is quickly becoming obsolete - so he’s decided to create a series of 7 podcasts, the firs...
New
PragmaticBookshelf
Quickly turn complex problems into simple, working solutions using functional programming, safe concurrency, and the expressive tools of ...
New
xiji2646-netizen
There’s a GitHub repo at forrestchang/andrej-karpathy-skills that’s sitting at 97.8k stars. It’s a single CLAUDE.md file with four behavi...
New