A knot is an embedding of a circle $S^{1}$ in $3$-dimensional Euclidean space, $\mathbb{R}^3$. Knots are considered equivalent under ambient isotopy. There are two different types of knots, *tame* and *wild*. A tame knot is any knot equivalent to a polygonal knot, that is a knot whose image in $\mathbb{R}^3$ is the union of a finite set of line segments. A wild knot is a knot that is not tame. In contrary to the tame knots, wild knots are not well-behaved creatures. They often provide counterexamples to the general version of those theorems which hold for the tame knots.

There are several invariants associated with the knots. For example, the crossing number of a knot is a knot invariant defined as the smallest number of crossings of any diagram of the knot. Wild knots (such as the following picture) may have infinite crossing numbers anywhere between $\aleph_0$ and $2^{\aleph_0}$.

**Remark 1.** Some cautiousness is needed in the previous statement. According to what Joel has mentioned in his comment, the last statement can be made more accurate. In fact, the set of crossing points of a wild knot *seems* to be projective which implies that assuming the existence of infinitely many Woodin cardinals, it is either countable or of size continuum because the axiom of Projective Continuum Hypothesis holds under this plausible large cardinal assumption. Furthermore, based on Andreas' comment, one may remove the large cardinal assumption in the last argument too.

While there has been an extensive amount of research dedicated to the classification of tame knots (particularly the iconic prime ones), up to ambient isotopy, it turned out that the classification of wild knots is generally much more complicated. (In this direction see Kulikov's paper: *A non-classification result for wild knots*).

My question simply is how to make the chaos caused by wild knots even worse by adding new ones to the universe through forcing.

Question 1.Assuming failure of Continuum Hypothesis, let $\aleph_0<\kappa\leq 2^{\aleph_0}$ be an uncountable cardinal. Is there a cardinal preserving forcing notion $\mathbb{P}$ of the universe $V$ such that in $V^{\mathbb{P}}$ there exists a wild knot $K$ of crossing number $\kappa$ which is different from all wild knots of crossing number $\kappa$ in the ground model up to ambient isotopy (definable in $V^{\mathbb{P}}$)?

**Remark 2.** Potentially, one may find some questions, related to the possible number/behavior of wild knots of the certain knot invariant (such as bridge, unknotting, stick numbers, etc.) in a forcing extension, interesting. However, for the sake of this question, I preferred to stick to a very specific one concerning crossing number. Please feel free to share your thoughts about other possibly interesting problems along these lines in the comments below.

**Update.** According to Ian's remark in the comments section, it turned out that the notion of crossing number for wild knots retains some ambiguity, as (in contrary to the case for tame knots) there is no straightforward theorem guaranteeing the existence of a *regular* projection for wild knots, which is somehow essential for counting the number of crossing points in a wild knot. Roughly speaking, if you have a wild knot, you can't even be sure how much knotted it is! Another indication of why these crazy creatures deserve the name "wild"!

Anyway, the soul of the original question remains valid even without any specification to the case of tame/wild knots of any particular characteristics in terms of knot invariants. The main problem is whether it is possible to add a morphologically new knot to the mathematical universe via forcing. Precisely:

Question 2.Starting from a model $V$ of $ZFC$, is there a (cardinal preserving) forcing notion $\mathbb{P}$ in $V$ such that in $V^{\mathbb{P}}$ there exists a (possibly wild) knot $K$ which is different from all knots of the ground model up to ambient isotopy (definable in $V^{\mathbb{P}}$)?