Certain connect sums of torus knots with infinitely many non-characterizing slopes

TL;DR

This paper demonstrates that certain connected sums of torus knots possess infinitely many slopes for which the resulting surgery manifolds do not uniquely identify the original knot, challenging the notion of characterizing slopes.

Contribution

It establishes that specific connected sums of torus knots have infinitely many non-characterizing slopes, expanding understanding of knot surgery characterization.

Findings

Certain connected sums of torus knots have infinitely many non-characterizing slopes.

Application of Baker and Motegi's condition to these knots.

Identification of non-unique surgery outcomes for these knots.

Abstract

For a knot $K,$ a slope $r$ is said to be characterizing if for no other knot $J$ does $r$-framed surgery along $J$ yield the same manifold as $r$-framed surgery on $K.$ Applying a condition of Baker and Motegi, we show that the knots $T_{2,2n+3}\#T_{-2,2n+1}$ have infinitely many non-characterizing slopes.