We give lower bounds for the Gordian distance and the unknotting number of handlebody-knots by using Alexander biquandle colorings. We construct handlebody-knots with Gordian distance n and unknotting number n for any positive integer n.
A spatial surface is a compact surface embedded in the 3-sphere. In this paper, we provide several typical examples of spatial surfaces and construct a coloring invariant to distinguish them. The coloring is defined by using a multiple group rack, which is a rack version of a multiple conjugation quandle.
We introduce a map, which we call the bind map, from the set of all classical braids to that of all trivalent braids. Then we define a notation for handlebody-links with a pair of a bind map and a classical braid. We show that isotopies and braid relations are equivalent for trivial handlebody-braids obtained from classical 3-braids with bind maps. We introduce two types of graphs for a classical braid, which indicate how the binding forms of the trivalent braids are transformed each other. We determine all patterns of the graphs for classical 3-braids.
We give lower bounds for the tunnel number of knots and handlebody-knots. We also give a lower bound for the cutting number, which is a "dual" notion to the tunnel number in handlebody-knot theory. We provide necessary conditions to be constituent handlebody-knots by using G-family of quandles colorings. The above two evaluations are obtained as the corollaries. Furthermore, we construct handlebody-knots with arbitrary tunnel number and cutting number.
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