# When can a link be obtained from another using crossing exchanges and smoothings?

## TL;DR

This paper investigates the computational complexity of transforming a link diagram into a fixed target link using crossing exchanges and smoothings, showing polynomial-time algorithms for certain classes of links.

## Contribution

It extends known polynomial-time algorithms to all torus links T_{2,m} and twist knots for the link transformation problem.

## Key findings

- Polynomial-time algorithms exist for T_{2,m} torus links.
- Polynomial-time algorithms exist for all twist knots.
- The problem remains computationally feasible for these classes.

## Abstract

Let $L$ be a fixed link. Given a link diagram $D$, is there a sequence of crossing exchanges and smoothings on $D$ that yields a diagram of $L$? We approach this problem from the computational complexity point of view. It follows from work by Endo, Itoh, and Taniyama that if $L$ is a prime link with crossing number at most $5$, then there is an algorithm that answers this question in polynomial time. We show that the same holds for all torus links $T_{2,m}$ and all twist knots.