The second Reidemeister move between a circle and an arc #
PDCode.insertCircleClasp D p b pushes a new crossing-free circle across the arc ending
at p, creating a cancelling pair of crossings. It is compared with D.adjoinCircle:
the circle becomes a crossing-bearing component, and the other components retain their
strands. Unlike clasp insertion between two existing arcs, there is no common-face
hypothesis: the new circle can be placed beside the chosen arc.
The new crossings have opposite over-pair indicators, so the same physical strand is
over at both crossings. The two internal clasp arcs join slots 2 to 1 and 3 to 0;
the outside arc of the circle joins slot 1 of the first crossing to slot 2 of the
second. The remaining ports attach to the cut arc. This is the circle-and-arc case;
clasp insertion between two crossing-free circles is a separate case.
The clasp has the same component count and Kauffman bracket as D.adjoinCircle, and
is planar exactly when D is planar. These results allow this local second Reidemeister
move within planar PD codes while preserving component count and the bracket; together
with the oriented move's writhe preservation, they give Jones polynomial invariance
for the circle-and-arc case.
References #
- W. B. R. Lickorish, An Introduction to Knot Theory, GTM 175 (1997), Chapter 1 and Chapter 3, Lemma 3.3 (the second Reidemeister move).
- L. H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), 395–407.
Push a new circle across an existing arc, creating a two-crossing Reidemeister clasp.
The circle is additional to the components of D; the result is compared to D.adjoinCircle.
The bit b selects the over-strand at the first new crossing.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The old crossing slots retain their half-edges.
The first new crossing uses the first four new half-edges.
The second new crossing uses the last four half-edges.
The arc ending at p enters the first crossing at slot 0.
The other end of the cut arc attaches to slot 3 of the second crossing.
Every half-edge off the cut arc keeps its old partner.
Partners of the first crossing's slots: the chosen arc, the outside circle arc, and the two internal clasp arcs.
Partners of the second crossing's slots.
Traversals and the four smoothings #
The four smoothings leave the original state circles, plus one circle exactly when
both local slot smoothings agree. The statement uses the actual A/B state choices.
Components and planarity #
Graph components correspond by retaining every old half-edge; both new crossings belong to the component containing the chosen arc.
Equations
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Instances For
The graph-component correspondence retains old representatives.
The inverse graph-component correspondence retains old representatives.
Every slot of the first new crossing maps back to the chosen arc's graph component.