Invariants of knots and links on \(T\)-polyhedra (Q5941876)

Let \(T\) denote the polyhedron \(\{(x,y,z)\in{\mathbb R}^3\mid z=0\text{ or } y=0,z\geq 0\}\). It is easy to show that any (tame) knot in \({\mathbb R}^3\) is isotopic to a simple closed curve in \(T\). This leaves open the question as to whether two knots in \(T\) may be isotopic as knots in \({\mathbb R}^3\) while being non-isotopic within \(T\). The paper defines an integer isotopy invariant \(\Theta(K)\) of knots \(K\) in \(T\), which is then used to demonstrate explicitly two knots in \(T\) which are isotopic in \({\mathbb R}^3\) but not in \(T\). The construction of \(\Theta\) uses a close analysis of the space (groupoid) \(\mathcal A\) of pairs of cycles in \(T\) with disjoint support, on which a family of isotopy invariants \(\omega_n: {\mathcal A}\rightarrow{\mathcal H}^n\) is defined, where \(\mathcal H\) denotes the Lie ring of the group \({\mathbb Z}^2*{\mathbb Z}^2\). Here \(\omega_1\) is essentially the linking pairing. For a knot \(K\) in \(T\), a quadratic form \(Q\) is defined on \(H_1(M_K,{\mathbb Z})\) (where \(M_K\) is a surface within \(T\) bounded by \(K\)) as the quadratic part of \(\omega_2(x,k)\) for a cycle \(k\) isotopic to \(K\). Then \(\Theta(K)\) is defined as the determinant of \(Q\).