A restriction on the Alexander polynomials of L-space knots

Abstract

Using an invariant defined by Rasmussen, we extend an argument given by Hedden and Watson which further restricts which Alexander polynomials can be realized by L-space knots.

On the Alexander polynomial of lens space knots

“…In the case of lens space knot in S 3 , this corollary was proven by the author in [14], although, here in more general cases we reprove this corollary by using the non-zero curve. It is proven in [5] and [3] that n 2 = g − 1 for any L-space knot in S 3 , where g is the genus of the knot.…”

“…Theorem 4.20. If a lens space knot K in an LZHS 3 satisfies 2g(K) − 4 ≤ k 2 ≤ 2g(K) − 2, then the lens surgery parameters are (11,3), (14,3), (19, 7) and are realized by T (3, 4), T (3, 5) or P r(−2, 3, 7),…”

“…If α(K) = 2 then p ≤ 10 and k 2 ≤ 3 by using Theorem 1.10. The surgery parameters with α(K) = 2 are (5, 2), (7, 2), (8,3) or (10,3). The half non-zero sequences of the parameters (8, 3) and (10, 3) are (4, 3, 1, 0) and (6, 5, 3, 2, 0) respectively.…”

“…Suppose that k 2 ≤ 3 (the second picture in Figure 11). From the inequality k ≤ k 2 , in this case, we have (p, k) = (5, 2), (8,3), or (10,3) only. Here the non-zero sequence of K 8,3 is N S h (K 8,3 ) = (4, 3, 1, 0) and for K 10,3 can be seen in Table 1.…”

“…Suppose that K is a lens space knot in an LZHS 3 with the parameters (26, 5), (24, 5), (13,5), (12,5), (17,4), (15,4), (19, 3), (17,3) and (10,3). Then some a i (K) (in Theorem 2.4) is not absolutely less than or equal to 1.…”