Kauffman-Harary conjecture holds for Montesinos Knots

A proof of the Kauffman-Harary Conjecture

We prove the Kauffman-Harary Conjecture, posed in 1999: given a reduced, alternating diagram D of a knot with prime determinant p, every non-trivial Fox p-coloring of D will assign different colors to different arcs.

Kauffman-Harary conjecture for alternating virtual knots

In 1999, Kauffman-Harary conjectured that every non-trivial Fox $p$-coloring of a reduced, alternating knot diagram with prime determinant $p$ is heterogeneous. Ten years later this conjecture was proved by W. Mattman and P. Solis. Mathew Williamson generalized this conjecture to alternating virtual knots and proved it for certain families of virtual knots. In the present note, we use the methods of W. Mattman and P. Solis to give an affirmative answer to the Kauffman-Harary conjecture for alternating virtual knots.

On the Harary-Kauffman Conjecture and Turk's Head Knots

The m,n Turk's Head Knot, THK(m,n), is an "alternating (m,n) torus knot." We prove the Harary-Kauffman conjecture for all THK(m,n) except for the case where m \geq 5 is odd and n \geq 3 is relatively prime to m. We also give evidence in support of the conjecture in that case. Our proof rests on the observation that none of these knots have prime determinant except for THK(m,2) when P_m is a Pell prime.

广义Kauffman-Harary猜想成立

The Generalized Kauffman-Harary Conjecture is True

本文证明了Asaeda、Przytycki和Sikora提出的广义Kauffman-Harary猜想：素链环的约化交错图中任意两条不同弧都可被某个Fox $δ$-着色区分。

4股蒙特西诺斯结外皮的卡夫曼括号 skein 模中的 torsion

Torsion in Kauffman bracket skein module of a $4$-strand Montesinos knot exterior

该研究证明了4股蒙特西诺斯结外皮的卡夫曼括号 skein 模存在torsion，回答了Kirby列表中的一个疑问。

5-moves and Montesinos links

We classify the Montesinos links up to mutation and 5-move equivalence, and obtain from this a Jones and Kauffman polynomial test for a Montesinos link.