Narrow Results

Two knots in three-space are S-equivalent if they are indistinguishable by Seifert matrices. We show that S-equivalence is generated by the doubled-delta move on knot diagrams. It follows as a corollary that a knot has trivial Alexander polynomial if and only if it can be undone by doubled-delta moves.

S-equivalence of classical knots is investigated, as well as its relationship with mutation and the unknotting number. Furthermore, we identify the kernel of Bredon's double suspension map, and give a geometric relation between slice and algebraically slice knots. Finally, we show that every knot is S-equivalent to a prime knot.

Our longterm plan is to classify knot modules and pairings by utilizing the power of computational number theory. The first step in this is to define invariants for which any given value arises from only finitely many modules: this is the purpose of the present paper.

In 2003, Naik and Stanford proved that two S-equivalent knots are related by a finite sequence of doubled-delta moves on their knot diagrams. We show that classical S-equivalence is not sufficient to extend their result to ordered links. We define a new algebraic relation on Seifert matrices, called strong S-equivalence, and prove that two oriented, ordered links L and L′ are related by a sequence of doubled-delta moves if and only if they are strongly S-equivalent. We also show that this is equivalent to the fact that L′ can be obtained from L through a sequence of Y-clasper surgeries, where each clasper leaf has total linking number zero with L.

Using Blanchfield pairings, we show that two Alexander polynomials cannot be realized by a pair of matrices with algebraic Gordian distance one if a corresponding quadratic equation does not have an integer solution. We also give an example of how our results help in calculating the Gordian distances, algebraic Gordian distances and polynomial distances.

For any virtual link $L=S\cup T$ that may be decomposed into a pair of oriented $n$-tangles $S$ and $T$, an oriented local move of type $T\mapsto \varphi (T)$ is a replacement of $T$ with the $n$-tangle $\varphi (T)$ in a way that preserves the orientation of $L$. After developing a general decomposition for the Jones polynomial of the virtual link $L=S\cup T$ in terms of various (modified) closures of $T$, we analyze the Jones polynomials of virtual links $L_1,L_2$ that differ via a local move of type $T\mapsto \varphi (T)$. Succinct divisibility conditions on $V(L_1)-V(L_2)$ are derived for broad classes of local moves that include the $\mathrm{\Delta }$-move and the double-$\mathrm{\Delta }$-move as special cases. As a consequence of our divisibility result for the double-$\mathrm{\Delta }$-move, we introduce a necessary condition for any pair of classical knots to be $S$-equivalent.