Signature, slicing foams, and crossing changes of Klein graphs
arxiv(2024)
摘要
A totally oriented Klein graph is a trivalent spatial graph in the 3-sphere
with a 3-coloring of its edges and an orientation on each bicolored link. A
totally oriented Klein foam is a 3-colored 2-complex in the 4-ball whose
boundary is a Klein foam and whose bicolored surfaces are oriented. We extend
Gille-Robert's signature for 3-Hamiltonian Klein graphs to all totally oriented
Klein graphs and develop an analogy of Murasugi's bounds relating the
signature, slice genus and unknotting number of knots. In particular, we show
that the signature of a totally oriented Klein graph produces a lower bound on
the negative orbifold Euler characteristic of certain totally oriented Klein
foams bounded by Γ. When Γ is abstractly planar, these negative
Euler characteristics, in turn, produce a lower bound on a certain natural
unknotting number for Γ. Mutatis mutandi, we produce lower bounds on the
corresponding Gordian distance between two totally oriented Klein graphs that
can be related by a sequence of crossing changes. We also give examples of
theta-curves for which our lower bounds on unknotting number improve on
previously known bounds.
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