arXiv research

We note that a rational $3$-tangle diagram is obtained from a combination of four generators. There is an algorithm to distinguish two rational $3$-tangle diagrams up to isotopy. However, there is no perfect classification about rational $3$-tangle diagrams such as the classification of rational $2$-tangle diagrams cor…

The paper defines normal forms for rational 3-tangles and shows a sequence of moves to transform one form to another.

Paper classifies rational 3-tangles using normal forms and minimal coordinates.

Simpler method detects trivial rational 3-tangle.

In this paper, We introduce an invariant of rational n-tangles which is obtained from the Kauffman bracket. It forms a vector with Laurent polynomial entries. We prove that the invariant classifies the rational 2-tangles and the reduced alternating rational 3-tangles. We conjecture that it classifies the rational 3-tan…

Proves a property of certain 3D knots and links.

A $3$-$tangle$ $T$ is the disjoint union of $3$ properly embedded arcs in the unit 3-ball; it is called rational if there is a homeomorphism of pairs from $(B^3,T)$ to $(D^2\times I,\{x_1,x_2,x_3\}\times I)$. Two rational 3-tangles $T$ and $T'$ are isotopic if there is an orientation-preserving self-homeomorphism $h: (…

The Kauffman bracket polynomial is calculated for specific Turk's head knots.

New formula recovers degree of colored Jones polynomials for pretzel knots.

Using the Hatcher-Oertel algorithm for finding boundary slopes of Montesinos knots, we prove the Slope Conjecture and the Strong Slope Conjecture for a family of 3-tangle pretzel knots. More precisely, we prove that the maximal degrees of the colored Jones polynomial of such knots determine a boundary slope as predicte…

Study on SU(2)-simple knots and cyclic 3-manifolds, extending known results.

Define web algebras for annular SL(2) and SL(3) using foam TQFTs.