By Francesco Costantino

We identify a calculus for branched spines of 3-manifolds through branched Matveev-Piergallini strikes and branched bubble-moves. We in short point out a few of its attainable purposes within the research and definition of State-Sum Quantum Invariants.

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4. 3 as a guide to write an argument that given any knot projection, the precrossings can be resolved to produce a diagram of the unknot. (Hint: There are two parts to this problem. First you must devise an algorithm to create the desired unknot diagram. ) Let K be a knot. 4, we can show that, given any diagram D for K, some number of crossings can be changed in D to produce a diagram of the unknot. Note that changing a crossing in a knot diagram is like passing the corresponding knot through itself in space.

What about when Ulysses (the unknotter) plays second? Formulate and prove two propositions about these cases. 2Pretzel Links Another fascinating collection of links is the family of pretzel links. These links are both rich in structure but also easy to visualize since they are made out of simple twists. 1. Let p, q, and r be integers. A 3-strand pretzel link Pp,q,r can be constructed as follows. Take three pairs of string segments and arrange them vertically. Twist the bottom ends of the first pair p times (in the counterclockwise direction if p > 0 and in the clockwise direction if p < 0).

1: Four concentric strands. 2: Examples of replaceable regions are shown in (a). The diagram (b) shows a nonexample of a replaceable region because it contains three strands. 2 (a). By construction, a closed n-braid diagram has an innermost region with the property that any line segment drawn from the innermost region to a point that is exterior to the link will intersect the diagram at exactly n points. We count both the under- and over-strands if the segment happens to pass through a crossing.

### A calculus for branched spines of 3-manifolds by Francesco Costantino

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