Problems in Knot Theory

There are already several collections of problems in knot theory available. So this is mainly a collection of problems of personal interest. Feel free to send me any comments you might have: maybe the answer is obvious, maybe the answer exists already in literature, maybe the problem should be attributed more appropriately, etc. Also, it is welcome if anyone would like to suggest some problems to this collection. Nevertheless, since this is a personal collection, I reserve the right to decide whether to put the suggested problems into this collection according to my own taste.

1. Surgery modification is a procedure of modifying a link in or by performing a Dehn surgery on an unknotted circle having linking number zero with all components of the link in question. Surgery equivalence is then the equivalence relation generated by surgery modification. The classification of links up to surgery equivalence is done by J. Levine (Topology,1987).

We may refine the notion of surgery modification as follows. For simplicity, let us consider only links with vanishing linking numbers. If we assume further that the unknotted circle used to perform a surgery modification has vanishing Milnor's triple linking numbers with other componenets of the given link, we call such a surgery modification of "second order". It can be shown that the Sato-Levine invariant is invariant under surgery modification of second order. Classify links with zero linking numbers up to surgery equivalence of second order.

2. Call a knot K n-adjacent to the unknot if for some diagram of K, we can pick n crossings so that switching any k>0 of these n crossings will change K to the unknot. For example, the trefoil and figure-8 are 2-adjacent to the unknot. And for any n, it is not hard to construct knots which are n-adjacent to the unknot. Is it true that every unknotting number 1 knot (which is 1-adjacent to the unknot) is 2-adjacent to the unknot? For example, is the following knot 2-adjacent to the unknot?

Remark: Ted Stanford just gave a nice solution to this problem: If a knot is 2-adjacent to the unknot, then its v_2 has to be equal to 1, 0 or -1. Since the knot in the picture above has v_2 = 2, it is not 2-adjacent to the unknot. But if we do have v_2 = 1, 0, or -1, how do we show that a knot is not 2-adjacent to the unknot? For example, is it true that a double knot is 2-adjacent to the unknot iff it is the trefoil or the figure eight knot?

Remark: It is shown in "Knot adjacency and satellites" ( Kalfagianni and Lin, TAIT, 138(2004), pp. 207--217) that 2-bridge knots of the form K_{p/q}, where p/q=[2q_1,2q_2] for some integers q_1,q_2, are precisely those knots that have genus one and are 2-adjacent to the unknot.

3. (Zhenghan Wang) Given a knot diagram universe with n double points. Inside the set R of resolutions of knots, there may be many unknots. Suppose K, K' both are unknots in R, is there a sequence of unknots in R such that and each is obtained from by switching exactly one crossing?

4. Suppose denotes a polynomial in one variable with complex coefficients. Then

The Mahler measure of F(x) is defined by the following formula:

Applying Jensen's Formula in complex analysis, we get

It is easy to see that we may define the Mahler measure for a Laurent polynomial similarly. For a nice introduction to the Mahler measure, see Graham Everest: "Measuring the height of a polynomial", The Mathematical Intelligencer, vol. 20, no. 3, 1998.

Let J(K) be the Jones polynomial of a knot K. Is it true that m(J(K))>0 if K is a non-trivial knot?

Remark: As observed by Oliver Dasbach, the roots of the Jones polynomial of the figure 8 knot all have norm 1. Therefore, we do have a non-trivial knot with m(J(K))=0. Does the achirality of the figure 8 knot plays role here?

5. Is Kontsevich's theorem (that every rational weight system is induced by a rational Vassiliev knot invariant) true for other fields, say, for integer mod 2?

6. (Feng Luo) Does every non-trivial knot group admit an irreducible representation into SL(2,R)?

7. Use Kontsevich integral to show the existence of a non-trivial knot with trivial Alexander-Conway polynomial. This might give us some hints to the problem of whether there is a non-trivial knot with trivial Jones polynomial.

8. What is the homotopy type of the space L(K) of long ropes (as shown in the picture below) with the fixed knot type K?

A conjecture would be that, if K is a prime knot, L(K) is homotopy equivalent to the circle iff K is non-trivial, with the fundamental group generated by the obvious loop in L(K) shown in the above picture. This question is motivated by a paper of Jacob Mostovoy: Short ropes and long knots. If the conjecture holds, the homotopy type of the space of short ropes studied by Mostovoy would be clear. A paper of Allen Hatcher "Spaces of knots" seems to be related with this problem.

9. Let {K} be the set of isotopy classes of unoriented knots in the 3-space. Two different elements in {K} are said to be adjacent if they differ by a single crossing change. Conjecture: If h: {K} ---> {K} is a bijection such that h(K) and h(K') are adjacent iff K and K' are adjacent, then h is either the identity or the mirror map. In other words, any bijection of {K} onto itself preserving the adjacency in both ways is induced by a diffeomorphism of the ambient space.

10. Is the knot signature the limit of a sequence of finite type knot invarinats?

.... more to come.