TY - JOUR

T1 - Detecting and visualizing 3-dimensional surgery

AU - Antoniou, Stathis

AU - Kauffman, Louis H.

AU - Lambropoulou, Sofia

PY - 2020/1/6

Y1 - 2020/1/6

N2 - Topological surgery in dimension 3 is intrinsically connected with the classification of 3-manifolds and with patterns of natural phenomena. In this, mostly expository, paper, we present two different approaches for understanding and visualizing the process of 3-dimensional surgery. In the first approach, we view the process in terms of its effect on the fundamental group. Namely, we present how 3-dimensional surgery alters the fundamental group of the initial manifold and present ways to calculate the fundamental group of the resulting manifold. We also point out how the fundamental group can detect the topological complexity of non-trivial embeddings that produce knotting. The second approach can only be applied for standard embeddings. For such cases, we give new visualizations of 3-dimensional surgery as rotations of the decompactified 2-sphere. Each rotation produces a different decomposition of the 3-sphere which corresponds to a different visualization of the 4-dimensional process of 3-dimensional surgery.

AB - Topological surgery in dimension 3 is intrinsically connected with the classification of 3-manifolds and with patterns of natural phenomena. In this, mostly expository, paper, we present two different approaches for understanding and visualizing the process of 3-dimensional surgery. In the first approach, we view the process in terms of its effect on the fundamental group. Namely, we present how 3-dimensional surgery alters the fundamental group of the initial manifold and present ways to calculate the fundamental group of the resulting manifold. We also point out how the fundamental group can detect the topological complexity of non-trivial embeddings that produce knotting. The second approach can only be applied for standard embeddings. For such cases, we give new visualizations of 3-dimensional surgery as rotations of the decompactified 2-sphere. Each rotation produces a different decomposition of the 3-sphere which corresponds to a different visualization of the 4-dimensional process of 3-dimensional surgery.

KW - 2-sphere

KW - 3-manifold

KW - 3-space

KW - 3-sphere

KW - blackboard framing

KW - decompactification

KW - framed surgery

KW - fundamental group

KW - handle

KW - knot group

KW - knots

KW - Poincaré sphere

KW - rotation

KW - stereographic projection

KW - surgery visualization

KW - topological process

KW - Topological surgery

KW - topology change

KW - torus

KW - TOPOLOGICAL SURGERY

KW - Poincare sphere

UR - http://www.scopus.com/inward/record.url?scp=85077874870&partnerID=8YFLogxK

U2 - 10.1142/S0218216519400157

DO - 10.1142/S0218216519400157

M3 - Article

AN - SCOPUS:85077874870

VL - 28

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

IS - 13

M1 - 1940015

ER -