# Union of knots and its Alexander polynomial

In 1957 Kinoshita and Terasaka in their article "On unions of knots" generalized the operation of connected sum; it's called $union$.

I have several questions :

1.) If the Alexander polynomial of a prime knot $K$ is divisible by the Alexander polynomials of other knots $K^{i}$, is it true that knot $K$ can be presented as a union of two knots $K^{i}$ and $K^{j}$ with the same Alexander polynomial (it is possible that $i=j$)?

2.) If a prime knot $K$ can be presented as a union of two knots $K^{i}$ and $K^{j}$ with different Alexander polynomials, is it true that the Alexander polynomial of $K$ will be prime?

Example for first question : It's true for knots $8_{5}, 8_{10}, 8_{15}, 8_{19}, 8_{20}, 8_{21}, 9_{16}, 9_{24}, 9_{28}$.

Alexander polynomials of all this knots have a common factor $t^2-t+1$ (its Alexander polynomial of $3_{1}$). And all of these knots can be presented as a union of two $3_{1}$s.

Example for second question : It's true for knots $9_{22}, 9_{25}, 9_{30}, 9_{36}, 9_{42}, 9_{43}, 9_{44}, 9_{45}$. Alexander polynomials of all this knots are prime. And all of these knots can be presented as a union of $4_1$ and $3_{1}$.

My problem

There is a knot $9_{38}$ with Alexander polynomial $5-14t+ 19t^2-14t^3+ 5t^4$ is divisable by $t^2-t+1$, but I can't find a presentation by a union of trefoils.

I will call minimal Alexander matrix of a knot the minimal square presentation matrix of Alexander module. Knot obtained by union of two knots with the same Alexander polynomial has a special form of minimal Alexander matrix.

And I see, that minimal Alexander matrix of $9_{38}$ can not be represented in this form . What does this mean?

Thank you for your attention! (KnotPlot image of $9_{38}$ added by J.O'Rourke.)

• (I tried to fix minor grammatical issues without changing the meaning. Roll back if I overreached.) – Joseph O'Rourke Jun 13 '15 at 14:06
• I tried to write this sentence in more detail. Has it become more clear? – Alina P. Jun 13 '15 at 15:35
• Yes, thanks. (Incidentally, I cannot answer any of your questions.) – Joseph O'Rourke Jun 13 '15 at 16:46
• I am afraid I have not much to contribute, but it would be nice if you could talk about the "special form of minimal Alexander matrix" of $K_1 \oplus K_2$ in relation to the minimal Alexander matrix of $K_1$ and $K_2$... then we can just read the intro of the paper and know what to do next – wilsonw Jun 18 '15 at 10:15
• The answer to the second question is negative. In short knot $12n_88$ can be presented like the union of $5_1$ and $3_1$, but it's Alexander polynomial is not prime. Later I will present a detailed explanation. – Alina P. Jun 27 '15 at 9:22