Tabulation of Prime Knots in Lens Spaces

Abstract

Using computational techniques we tabulate prime knots up to five crossings in the solid torus and the infinite family of lens spaces L(p, q). For these knots we calculate the second and third skein module and establish which prime knots in the solid torus are amphichiral. Most knots are distinguished by the skein modules. For the handful of cases where the skein modules fail to detect inequivalent knots, we calculate and compare the hyperbolic structures of the knot complements. We were unable to resolve a handful of 5-crossing cases for p ≥ 13.

Knot Invariants in Lens Spaces

“…It is shown in [11] that 526 and 527 are non-isotopic in any lens space L(p, 1), but due to the symmetric nature of the two knots, none of our invariants are able to detect this. p S2,∞(526) = S2,∞(527) 2…”

“…We finish by presenting some explicit calculations of difficult cases of links in L(p, 1) where the mentioned invariants fail to detect inequivalent links. The knot notations are taken from the lens space knot table constructed in [11]. The Kauffman bracket skein modules and HOMFLY-PT skein modules (evaluated in the standard basis) were computed by the C++ program available in [10] (the algorithm itself is presented [11]).…”

Algebraic links in lens spaces

“…Links in L p,q . Knots and links in lens spaces were studied in [3,5,11]. We briefly recall some basic notions which will be used in the paper.…”

“…Indeed: consider the knot 3 7 in L 2,1 , that is given by diagrams in Figure 6. In lens space L 2,1 , the knot 3 7 is equivalent to the knot 2 1 (see Figure 8) by [5,Appendix B]. Applying Proposition 2.3, we can see that the lift of 3 7 in the 3-sphere is equivalent to the torus link T (8, 2).…”

The Alexander polynomial of links in lens spaces

“…The Alexander polynomial is indeed a useful invariant that can detect inequivalent knots that are not distinguishable by other common invariants. Let K 1 and K 2 be knots in L(3, 1) with diagrams from Figure 7, which are respectively the knots 1 1 and 5 1 from the knot Table 4 in [8] and Appendix D in [9]), since it holds that…”