Abstract:
We introduce a notion of natural orderings of elements of finite connected quandles of order $n$. When the elements of such a quandle $Q$ are already ordered naturally, any automophism on $Q$ is a natural ordering. Although there are many natural orderings, the operation tables for such orderings coincide when the permutation $*q$ is a cycle of length $n-1$. This leads to the classification of automorphisms on such a quandle. Moreover, it is also shown that every row and column of the operation table of such a quandle contains all the elements of $Q$, which is due to K. Oshiro. We also consider the general case of finite connected quandles.

Abstract:
Using unknotting number, we introduce a link diagram invariant of Hass and Nowik type, which changes at most by 2 under a Reidemeister move. As an application, we show that a certain infinite sequence of diagrams of the trivial two-component link need quadratic number of Reidemeister moves for being unknotted with respect to the number of crossings. Assuming a certain conjecture on unknotting numbers of a certain series of composites of torus knots, we show that the above diagrams need quadratic number of Reidemeister moves for being splitted.

Abstract:
In the previous paper, we considered a link diagram invariant of Hass and Nowik type using regular smoothing and unknotting number, to estimate the number of Reidemeister moves needed for unlinking. In this paper, we introduce a new link diagram invariant using irregular smoothing, and give an example of a knot diagram of the unknot for which the new invariant gives a better estimation than the old one.

Abstract:
Let $D(p,q)$ be the usual knot diagram of the $(p,q)$-torus knot, that is, $D(p,q)$ is the closure of the $p$-braid $(\sigma_1^{-1} \sigma_2^{-1}... \sigma_{p-1}^{-1})^q$. As is well-known, $D(p,q)$ and $D(q,p)$ represent the same knot. It is shown that $D(n+1,n)$ can be deformed to $D(n,n+1)$ by a sequence of $\{(n-1)n(2n-1)/6 \} + 1$ Reidemeister moves, which consists of a single RI move and $(n-1)n(2n-1)/6$ RIII moves. Using cowrithe, we show that this sequence is minimal over all sequences which bring $D(n+1,n)$ to $D(n,n+1)$.

Abstract:
A knot K is called a 1-genus 1-bridge knot in a 3-manifold M if (M,K) has a Heegaard splitting (V_1,t_1)\cup (V_2,t_2) where V_i is a solid torus and t_i is a boundary parallel arc properly embedded in V_i. If the exterior of a knot has a genus 2 Heegaard splitting, we say that the knot has an unknotting tunnel. Naturally the exterior of a 1-genus 1-bridge knot K allows a genus 2 Heegaard splitting, i.e., K has an unknotting tunnel. But, in general, there are unknotting tunnels which are not derived form this procedure. Some of them may be levelled with the torus \partial V_1=\partial V_2, whose case was studied in our previous paper. In this paper, we consider the remaining case.

Abstract:
A knot K in a closed connected orientable 3-manifold M is called a 1-genus 1-bridge knot if (M,K) has a splitting into two pairs of a solid torus V_i (i=1,2) and a boundary parallel arc in it. The splitting induces a genus two Heegaard splitting of the exterior of K naturally, i.e., K has an unknotting tunnel. However the converse is not true in general. Then we study such general case in this paper. One of the conclusions is that the unknotting tunnel may be levelled with the torus \partial V_1=\partial V_2.

Abstract:
If a rectangular diagram represents the trivial knot, then it can be deformed into the rectangular diagram with only two vertical edges by a finite sequence of merge operations and exchange operations, without increasing the number of vertical edges, which was shown by I. A. Dynnikov. We show in this paper that we need no merge operations to deform a rectangular diagram of the trivial knot to one with no crossings.

Abstract:
We determine all the Q-fundamental surfaces in $(p,1)$-lens spaces and $(p,2)$-lens spaces with respect to natural triangulations with $p$ tetrahedra. For general $(p,q)$-lens spaces, we give an upper bound for elements of vectors which represent Q-fundamental surfaces with no quadrilateral normal disks disjoint from the core circles of lens spaces. We also give some examples of non-orientable closed surfaces which are Q-fundamental surfaces with such quadrilateral normal disks.

Abstract:
We give a concrete example of an infinite sequence of $(p_n, q_n)$-lens spaces $L(p_n, q_n)$ with natural triangulations $T(p_n, q_n)$ with $p_n$ taterahedra such that $L(p_n, q_n)$ contains a certain non-orientable closed surface which is fundamental with respect to $T(p_n, q_n)$ and of minimal crosscap number among all closed non-orientable surfaces in $L(p_n, q_n)$ and has $n-2$ parallel sheets of normal disks of a quadrilateral type disjoint from the pair of core circles of $L(p_n, q_n)$. Actually, we can set $p_0=0, q_0=1, p_{k+1}=3p_k+2q_k$ and $q_{k+1}=p_k+q_k$.

Abstract:
If a knot has the Alexander polynomial not equal to 1, then it is linear $n$-colorable. By means of such a coloring, such a knot is given an upper bound for the minimal quandle order, i.e., the minimal order of a quandle with which the knot is quandle colorable. For twist knots, we study the minimal quandle orders in detail.