Algorithm for computing the Arf invariant of a knot According to "The knot book", by Colin Adams, two knots are pass equivalent if they are related by a finite sequence of pass-moves. Moreover every knot is pass-equivalent to either the unknot or the trefoil knot and these two knots are not pass-equivalent. 

We define Arf invariant of a knot to be 0 if the know is pass equivalent to unknot and to be 1 if it is pass equivalent to the trefoil knot.
Now here is my question: I am not interested to know how a knot is pass-equivalent to unknot or the trefoil knot. I just want to know if there is an easy algorithm for computing the Arf invariant of a knot based on its projection (for example, like the algorithm that we use to compute the linking number of links)? 
Can we move along a knot and compute its Arf invariant by taking its crossings somehow into consideration (maybe by counting its positive and negative crossings)?
 A: The well-known (see here) relations between the Arf invariant and the Alexander/Jones polynomials give you algorithms for computing the Arf invariant, if that's all you want.

But it sounds like what you're after is a 'crossing-linear' algorithm, in the sense that we go along the knot diagram, examine each crossing once, update some internal state, and at the end spit out the Arf invariant.
I don't know of such an algorithm.  But it's not terrifically hard to come up with a close approximation to this.
We'll use the property mentioned in the above link (there's a proof Kauffman's Formal Knot Theory, where this is cited as a folklore theorem) that for a standard smoothing $K_+, K_-, K_0$ of a crossing, where $K_0 = L_1 \cup L_2$ is a two-component link, we have $$\mathrm{Arf}(K_+) + \mathrm{Arf}(K_-) \equiv \mathrm{lk}(L_1, L_2) \mod{2}.$$
We can definitely compute the right-hand side by passing along the crossings of $K_0$, so all that remains is to choose crossings cleverly so that whichever one of $K_+, K_-$ is not our original knot is a bit simpler.  (Alternately, if you're happy with an algorithm that's exponential-time in the number of crossings you could do some state-sum-style approach here.)  This territory is a bit more well-known; I'd look at the more efficient algorithms for the Jones polynomial implemented by Bar-Natan in his Knot Theory package for Mathematica.  Ultimately this should get you something that's 'crossing-polynomial', maybe even 'crossing-quadratic'.
A: Arf invariant is modulo two reduction of the Vassiliev invariant $v_2$ (the coefficient of $z^2$ in the Conway polynomial). Thus a well-known Gauss diagram formula for $v_2$ can be used for its computation. Namely, fix a knot diagram  and choose a base point distinct from the crossings. Traverse the diagram starting from the base point. Each crossing is visited twice; write down the sequence in which we visit these crossings, denoting a passage on an overpass by $O_i$ and on an underpass by $U_i$. The result is the Gauss code of the knot diagram. For example, the standard diagram of a trefoil will be encoded by $U_1 O_2 U_3 O_1 U_2 O_3$. Then the Arf invariant is the number of "linked" pairs of crossings $...U_i...O_k...O_i...U_k...$ in the Gauss code.
A: This is too long for a comment, it's just an idea for this computation. Start with a Seifert surface $F \subset \Bbb R^3$ for the knot $K$. Up to isotopy, we can assume that $F$ projects regularly to the plane $\Bbb R^2 \subset \Bbb R^3$. Now, $F$ can be presented by bands attached to a disk. Up to pass-moves, we can unlink and unknot the bands of $F$, as these moves correspond to Reidemeister moves applied to the core graph of $F$ with its band decomposition. Now, the bands are trivial but can be twisted. By further pass-moves we can cancel any two full twists in any band, giving a new surface $F'$ such that any band has 0 or 1 full twists. So, the Arf invariant should be the total number of the full twists of $F'$ (mod 2).
I don't have a proof now, it's just intuition, the motivation being that the pass-moves preserve the Arf invariant.
By the way, this should imply that the Arf invariant is the total twisting of the bands of the original $F$ (mod 2).
