The Classification of Spun Torus Knots

Abstract

S. Satoh has defined a construction to obtain a ribbon torus knot given a welded knot. This construction is known to be surjective. We show that it is not injective. Using the invariant of the peripheral structure, it is possible to provide a restriction on this failure of injectivity. In particular we also provide an algebraic classification of the construction when restricted to classical knots, where it is equivalent to the torus spinning construction.

Some Polynomial Invariants of Welded Links

“…By the similar computation, in the case of outside connection [15][26][34], we have Q D T (t) − Q D P (t) = (t − 1) 2 A(t) for some A(t) ∈ Z[t ±1 ]. Similarly, we obtain the same result for the other cases of T and P with different orientations.…”

“…B. Winter [15] gave a specific example of inequivalent welded knots which are mapped to the same ribbon torus knot by the Tube operation, and he also showed that if two classical knot diagrams are welded equivalent, then they are equivalent as classical knots as well. It is worth noting that the knot group and Alexander polynomial are welded isotopy invariant and so any knot with non-trivial knot group or non-trivial Alexander polynomial is not welded isotopic to the unknot.…”

Balloons and Hoops and their Universal Finite-Type Invariant, BF Theory, and an Ultimate Alexander Invariant

“…Thus we have the following: Theorem 2.7 When restricted to long knots (which are the same as knots), δ is injective. Remark 2.8 A similar map studied by Winter [29] is (sometimes) 2 to 1, as it retains less orientation information.…”

“…Equation (29) amounts to writing the group law of a 2D Lie group in terms of its 2D Lie algebra, L 0 := span(α, β), and this is again an exercise in 2 × 2 matrix algebra, though a slightly harder one. We work in the adjoint representation of L 0 and aim to compare the exponential of the left hand side of (29) with the exponential of its right hand side.…”

Ribbon Torus Knots Presented by Virtual Knots With Up to Four Crossings

“…The terminology of vertical and horizontal mirror images follows Green [5]. The notations D * and D † are different from those used in [8,20,27,30].…”