On the character variety of group representations in SL (2, C) and PSL (2, C).
Habiro gave principal ideals of in which certain linear combinations of the colored Jones polynomials of algebraically-split links take values. The author proved that the same linear combinations for ribbon links, boundary links and Brunnian links are contained in smaller ideals of generated by several elements. In this paper, we prove that these ideals also are principal, each generated by a product of cyclotomic polynomials.
We define a measure of “complexity” of a braid which is natural with respect to both an algebraic and a geometric point of view. Algebraically, we modify the standard notion of the length of a braid by introducing generators , which are Garside-like half-twists involving strings through , and by counting powered generators as instead of simply . The geometrical complexity is some natural measure of the amount of distortion of the times punctured disk caused by a homeomorphism. Our main...
The Kauffman-Harary conjecture is a conjecture for Fox's colorings of alternating knots with prime determinants. We consider a conjecture for Alexander quandle colorings by referring to the Kauffman-Harary conjecture. We prove that this new conjecture is true for twist knots.
We study properties of the signature function of the torus knot . First we provide a very elementary proof of the formula for the integral of the signature over the circle. We also obtain a closed formula for the Tristram-Levine signature of a torus knot in terms of Dedekind sums.