Untangle problems, with knot theory

“Untangle problems, with knot theory” offers a basic introduction to the mathematical subfield of knot theory, including the classification of knots by crossing numbers. A mathematical knot is a closed loop that may or may not be tangled. Two knots are considered the same if one may be manipulated into the other using easy-to-understand techniques. Readers learn to identify knots by crossing numbers and encounter numerous hand-drawn sketches of knots, including the trivial knot, trefoil knot, figure-eight knot, and more. Mathematics students and enthusiasts are encouraged to employ knot theory methods for untangling problems in mathematics or life by asking whether they have encountered the problem before. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Group Geometrical Axioms for Magic States of Quantum Computing

Let H be a nontrivial subgroup of index d of a free group G and N be the normal closure of H in G. The coset organization in a subgroup H of G provides a group P of permutation gates whose common eigenstates are either stabilizer states of the Pauli group or magic states for universal quantum computing. A subset of magic states consists of states associated to minimal informationally complete measurements, called MIC states. It is shown that, in most cases, the existence of a MIC state entails the two conditions (i) N = G and (ii) no geometry (a triple of cosets cannot produce equal pairwise stabilizer subgroups) or that these conditions are both not satisfied. Our claim is verified by defining the low dimensional MIC states from subgroups of the fundamental group G = π 1 ( M ) of some manifolds encountered in our recent papers, e.g., the 3-manifolds attached to the trefoil knot and the figure-eight knot, and the 4-manifolds defined by 0-surgery of them. Exceptions to the aforementioned rule are classified in terms of geometric contextuality (which occurs when cosets on a line of the geometry do not all mutually commute).

2-Adjacency between some special knots and pretzel knot

We study 2-adjacency between a classical (3-strand) pretzel knot and the trefoil knot or the figure-eight knot by using the early results about classical pretzel knots and their polynomials and elementary number theory. We show that except for the trefoil knot or the figure-eight knot, a nontrivial classical pretzel knot is not 2-adjacent to either of them, and vice versa.

The L2-Alexander invariant is stronger than the genus and the simplicial volume

We study how the genus, the simplicial volume and the [Formula: see text]-Alexander invariant of Li and Zhang can detect individual knots among all others. In particular, we use various techniques coming from hyperbolic geometry and topology to prove that the [Formula: see text]-Alexander invariant contains strictly more information than the pair (genus, simplicial volume). Along the way, we prove that the [Formula: see text]-Alexander invariant detects the figure-eight knot [Formula: see text], the twist knot [Formula: see text] and an infinite family of cables on the figure-eight knot.