Intrinsic symmetry groups of links with 8 and fewer crossings

• Published in: arXiv.org 2011-09
• Main Authors: Berglund, Michael, Cantarella, Jason, Meredith Perrie Casey, Dannenberg, Ellie, Whitney, George, Johnson, Aja, Kelly, Amelia, LaPointe, Al, Mastin, Matt, Parsley, Jason, Rooney, Jacob, Whitaker, Rachel
• Format: Article
• Language: English
• Subjects: Group theory; Homomorphisms; Knots; Links; Mapping; Polynomials; Symmetry
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1010.3234

We present an elementary derivation of the "intrinsic" symmetry groups for knots and links of 8 or fewer crossings. The standard symmetry group for a link is the mapping class group \(\MCG(S^3,L)\) or \(\Sym(L)\) of the pair \((S^3,L)\). Elements in this symmetry group can (and often do) fix the link and act nontrivially only on its complement. We ignore such elements and focus on the "intrinsic" symmetry group of a link, defined to be the image \(\Sigma(L)\) of the natural homomorphism \(\MCG(S^3,L) \rightarrow \MCG(S^3) \cross \MCG(L)\). This different symmetry group, first defined by Whitten in 1969, records directly whether \(L\) is isotopic to a link \(L'\) obtained from \(L\) by permuting components or reversing orientations. For hyperbolic links both \(\Sym(L)\) and \(\Sigma(L)\) can be obtained using the output of \texttt{SnapPea}, but this proof does not give any hints about how to actually construct isotopies realizing \(\Sigma(L)\). We show that standard invariants are enough to rule out all the isotopies outside \(\Sigma(L)\) for all links except \(7^2_6\), \(8^2_{13}\) and \(8^3_5\) where an additional construction is needed to use the Jones polynomial to rule out "component exchange" symmetries. On the other hand, we present explicit isotopies starting with the positions in Cerf's table of oriented links which generate \(\Sigma(L)\) for each link in our table. Our approach gives a constructive proof of the \(\Sigma(L)\) groups.