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For each prime p > 7 we obtain the expression for an upper bound on the minimum number of colors needed to non-trivially color T(2, p), the torus knot of type (2, p), modulo p. This expression is t + 2l -1 where t and l are extracted from the prime p. It is obtained from iterating the so-called Teneva transformations which we introduced in a previous article. With the aid of our estimate we show that the ratio "number of colors needed vs. number of colors available" tends to decrease with increasing modulus p. For instance as of prime 331, the number of colors needed is already one tenth of the number of colors available. Furthermore, we prove that 5 is the minimum number of colors needed to non-trivially color T(2, 11) modulo 11. Finally, as a preview of our future work, we prove that 5 is the minimum number of colors modulo 11 for two rational knots with determinant 11.

We will discuss a method for visual presentation of knotted surfaces in the four space, by examining a number and a position of its Morse’s critical points. Using this method, we will investigate surface-knot with one critical point of index $1$. Then we show infinitely many mutually distinct surface-knots that have an embedding with two critical points of index $1$. Next we define a long flat form of a banded link for any surface-knot and show diagrammatically a long flat form of $n$-twist-spun $(2,t)$-torus knots.

We call a delta diagram any diagram of a knot or link whose regions (including the unbounded one) have 3, 4, or 5 sides. We prove that any knot or link admits a delta diagram. We define and estimate combinatorial link invariants stemming from this definition.

In this paper, we describe a two-dimensional rectangular-cell-complex derived from a surface-knot diagram of a surface-knot. We define a pseudo-cycle for a quandle colored surface-knot diagram. We show that the maximal number of pseudo-cycles is a surface-knot invariant.

It is known that a knot complement (minus two points) decomposes into ideal octahedra with respect to a given knot diagram. In this paper, we study the Ptolemy variety for such an octahedral decomposition in perspective of Thurston’s gluing equation variety. More precisely, we compute explicit Ptolemy coordinates in terms of segment and region variables, the coordinates of the gluing equation variety motivated from the volume conjecture. As a consequence, we present an explicit formula for computing the obstruction to lifting a boundary-parabolic $\mathrm{\text{PSL}}(2,ℂ)$-representation to boundary-unipotent $\mathrm{\text{SL}}(2,ℂ)$-representation. We also present a diagrammatic algorithm to compute a holonomy representation of the knot group.