What allowed two knot diagrams to represent the same mathematical knot?
Answer
Continuous deformation via Reidemeister moves
In the topological framework used by modern knot theory, two knot diagrams are considered mathematically identical if one can be transformed into the other through a specific set of allowable manipulations. These crucial manipulations are formally known as Reidemeister moves. These moves represent the fundamental ways a knot diagram can be altered continuously in space without introducing tears or gluing new connections. Recognizing that a sequence of these moves connects two diagrams confirms they belong to the same equivalence class, signifying they represent the same underlying mathematical knot structure, regardless of how complex or differently drawn the initial diagrams might appear.
Related Questions
What motivated Lord Kelvin and Peter Guthrie Tait's study of knots?Knot theory emerged as a branch of which abstract mathematical field?What mathematical descriptors are sought to distinguish non-equivalent knots?What allowed two knot diagrams to represent the same mathematical knot?What significant leap in knot theory followed the initial attempts by Thomson and Tait?Which physicist connected quantum field theories to topological invariants like the Jones polynomial?What distinguishes the analytical "knot system" from prehistory knot tying?What specific area of study was Johann Benedict Listing noted for in early topology?What was the immediate purpose of cataloging knot configurations by Kelvin and Tait?Besides quantum gravity, what current area applies knot mathematics to physical structures?