There exist many constructions of infinite families of knots with the same Alexander polynomial. However, alternating knots seem very special. While there are also many result on restricting the form the Alexander polynomial of an alternating knot can have, is it known whether infinitely many alternating knots can have the same Alexander polynomial?

1$\begingroup$ Why the heegaardfloerhomology tag? $\endgroup$– HJRWOct 27, 2014 at 10:36

$\begingroup$ @HJRW: Because Heegaard Floer homology is very much linked to Alexander polynomial (Alexander polynomial is the Euler characteristic of the knot Floer homology) and some results restricting Alexander polynomial of alternating knots were obtained using Heegaard Floer homology (e.g. arxiv.org/abs/math/0209149 ). Also, I think it might be interesting to people doing Heegaard Floer homology. $\endgroup$– shestipalovOct 27, 2014 at 11:20

1$\begingroup$ The KinoshitaTerasaka knot is alternating and has a trivial alexander polynomial. So the connectsum of this knot with itself (arbitrary many times) is an infinite family of knots with trivial Alexander polynomial and these have alternating diagrams. $\endgroup$– Ryan BudneyOct 27, 2014 at 14:29

2$\begingroup$ No Ryan, KinoshitaTerasaka is not alternating. $\endgroup$– Ian AgolOct 27, 2014 at 17:25

$\begingroup$ Whoops, I thought it was. $\endgroup$– Ryan BudneyOct 27, 2014 at 21:06
1 Answer
No, there cannot exist infinitely many alternating knots with the same Alexander polynomial.
To see why, suppose for the contrary that $K$ belongs to an infinite family $\{K_n\}_{n\in\mathbb{Z}}$ of alternating knots with $\Delta_{K_n}(t)=\Delta_K(t)$. Immediately we have $$ \det(K_n ) = \Delta_{K_n} (−1) = \Delta_K (−1) = \det(K ). $$ for all $n$. Because these knots are alternating, each one admits a reduced alternating diagram, say $D_n$ with crossing number $c(D_n)$. The Bankwitz Theorem implies that $c(K_n) \leq \det(K_n) = \det(K)$. However, there are only finitely many knots of a given crossing number, and in particular $c(K_n)$ grows arbitrarily large with $n$, which contradicts that $c(K_n) \leq \det(K)$.
For this reason, there also cannot exist infinitely many alternating knots with the isomorphic (bigraded) $\widehat{HFK}(S^3, K)$, either.
In general though, I think these types of botany questions for the Alexander polynomial are interesting both on their own, and in the context of knot Floer homology. Some nice open botany and geography questions were recently mentioned in a paper of Hedden and Watson (see http://arxiv.org/abs/1404.6913).

1$\begingroup$ Out of curiosity, why do they use the word "botany" in this context? $\endgroup$– N. OwadOct 28, 2014 at 18:12

2$\begingroup$ In studying some topological invariants, people use the word
geography' to ask what range of invariants are realized by spaces of a given character. Eg what pairs (Euler characteristic, signature) are realized by minimal complex surfaces.
Botany' refers to how many there are with given invariants. Eg, how many smooth structures on a given complex surface. The analogy would be, say, how many roses [species of flower in genus rosa] are there. $\endgroup$ Nov 1, 2014 at 3:58