Hyperbolic (1,2)-knots in S^3 with crosscap number two and tunnel number one

TL;DR

This paper introduces a method to construct specific hyperbolic knots in S^3 with crosscap number two and tunnel number one, expanding the known classes of such knots beyond 2-bridge and (1,1)-knots.

Contribution

It provides a novel construction technique for hyperbolic (1,2)-knots with crosscap number two and tunnel number one, including an explicit infinite family.

Findings

Constructed a new class of hyperbolic (1,2)-knots with specified properties.

Demonstrated these knots are neither 2-bridge nor (1,1)-knots.

Provided detailed examples of an infinite family of such knots.

Abstract

A knot in S^3 is said to have crosscap number two if it bounds a once-punctured Klein bottle but not a Moebius band. In this paper we give a method of constructing crosscap number two hyperbolic (1,2)-knots with tunnel number one which are neither 2-bridge nor (1,1)-knots. An explicit infinite family of such knots is discussed in detail.