Skip to content

Use Case

As described by the notation for weighted planar tangle trees (WPTT) we may linearize a WPTT with a modified balanced parentheses tree notation strategy 1.

Definition

For a given WPTT linearization we construct the pure vignette for the WPTT by:

  1. expanding all sticks and ring subtrees with parentheses
  2. dropping all weights

Theorem

The pure vignette of an Right Leaning Identity Tangle Tree (RLITT) is invariant.

Proof

Bonahon and Siebenmann 2 tell us that the abstract tree underlying a weighted planar tree is invariant (consequence of proposition 12.22). That is, two weighted planar trees resolve to the same knot type if and only if there is a tree isomorphism between their underlying abstract trees. This extends in the obvious way to WPTT with underlying rooted plane trees. Since RLITT are unique representatives of the isotopy class we have the desired result.

Example

On the top we find an arborescent tangle with its tree and linearization. On the bottom we find the rooted plane tree as well as the pure vignette associated with the tangle.

Example

On the top we find a rational tangle with its tree and linearization. On the bottom we find the rooted plane tree as well as the pure vignette associated with the tangle.

  1. J. Ian Munro and Venkatesh Raman. Succinct Representation of Balanced Parentheses and Static Trees. SIAM Journal on Computing, 31(3):762–776, 2001. URL: http://epubs.siam.org/doi/10.1137/S0097539799364092 (visited on 2025-12-29), doi:10.1137/S0097539799364092. ↩

  2. Francis Bonahon and Laurence C Siebenmann. New Geometric Splittings of Classical Knots and the Classification and Symmetries of Arborescent Knots. 2016. URL: https://dornsife.usc.edu/francis-bonahon/wp-content/uploads/sites/205/2023/06/BonSieb-compressed.pdf. ↩