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Suppose $L$ is an oriented link and $L'$ is a link obtained from the Stallings' twists.

Are there any functoriality of the twist operation on the links, such as functoriality between (1) homology groups $H_*(S^3 - L)$, $H_*(S^3 - L')$ of link complements, (2) Thurston norm or Alexander norm, or (3) Teichmuller polynomials if those links are fibered.

Any relation or induced maps are welcome even it is not functorial at all..

Thank you in advance.

EDIT--- The Stallings' twist in my settings is, if there is a link $L$ then choose a curve $C$ which is an unknot in $S^3$ and the linking number between $L$ is $0$. Now thicken $C$ and we get a solid torus whose core is $C$. The Dehn surgery of slope $\pm1$ along this torus is what I call "Stallings' twist".

For example, If $L$ is a torus link $(2, 2)$, which is in fact a Hopf link, then choose $C$ as the meridian of the torus. Taking the twist $n$ times will yields a $(2, 2n+2)$ torus link.

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  • $\begingroup$ Could you tell us your definition of "Stallings' twists"? $\endgroup$
    – Ryan Budney
    Commented Feb 6, 2023 at 14:26
  • $\begingroup$ While you're at it, perhaps you could say what you mean by `functoriality' in this setting. $\endgroup$
    – Danny Ruberman
    Commented Feb 6, 2023 at 19:04
  • $\begingroup$ An orientation on a link determines a basis for the first homology; orientations are preserved by Stallings twists. So in that sense the homology groups you asked about are canonically identified. $\endgroup$
    – Danny Ruberman
    Commented Feb 6, 2023 at 19:04
  • $\begingroup$ Dear Ryan, I just editted my settings. If I understand it incorrectly please let me know. sorry for omitting the details. $\endgroup$
    – jhbaik
    Commented Feb 7, 2023 at 1:26
  • $\begingroup$ Dear Danny, What I mean by functoriality is that after some operation the link changes, and I wonder if one can induce some maps on homology or Teichmuller polynomials or some other link invariants from the twisting operation. $\endgroup$
    – jhbaik
    Commented Feb 7, 2023 at 1:28

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