NP-hard problems naturally arising in knot theory

Abstract

We prove that certain problems naturally arising in knot theory are NP--hard or NP--complete. These are the problems of obtaining one diagram from another one of a link in a bounded number of Reidemeister moves, determining whether a link has an unlinking or splitting number $k$, finding a $k$-component unlink as a sublink, and finding a $k$-component alternating sublink.

Figures (11)

• Figure 1: Hopf links, one for each variable $x_i$.
• Figure 2: Dividing the box into m shorter boxes vertically, to put each clause component in.
• Figure 3: A link representing the commutator $[\neg x_1,[\neg x_3,x_4]]$, corresponding to the clause $\neg x_1 \vee \neg x_3 \vee x_4$.
• Figure 4: An example of an unlinking move replacing (a) with (b).
• Figure 5: (a) A Hopf link and (b) the link obtained by replacing each component with a Whitehead double.

Theorems & Definitions (26)

• Theorem 1
• proof
• Definition 2
• Theorem 3
• Lemma 4
• proof
• proof : Proof of Theorem \ref{['thm_unlinking']}
• Corollary 5
• proof
• Remark 6