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Quandles with involutions that satisfy certain conditions, called good involutions, can be used to color non-orientable surface-knots. We use subgroups of signed permutation matrices to construct non-trivial good involutions on extensions of odd order dihedral quandles. For the smallest example of order 6 that is an extension of the three-element dihedral quandle, various symmetric quandle homology groups are computed, and applications to the minimal triple point number of surface-knots are given.
Motivated by the construction of free quandles and Dehn quandles of orientable surfaces, we introduce Dehn quandles of groups with respect to their subsets. As a characterisation, we prove that Dehn quandles are precisely those quandles which embed n
We investigate the complexity of finding an embedded non-orientable surface of Euler genus $g$ in a triangulated $3$-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance of embeddab
A longstanding avenue of research in orientable surface topology is to create and enumerate collections of curves in surfaces with certain intersection properties. We look for similar collections of curves in non-orientable surfaces. A surface is non
A homology theory is developed for set-theoretic Yang-Baxter equations, and knot invariants are constructed by generalized colorings by biquandles and Yang-Baxter cocycles.
By considering negative surgeries on a knot $K$ in $S^3$, we derive a lower bound to the non-orientable slice genus $gamma_4(K)$ in terms of the signature $sigma(K)$ and the concordance invariants $V_i(overline{K})$, which strengthens a previous boun