In knot theory, splicing generally has more than just one or two inputs.

Splicing with one input generates things like Whitehead doubles: Whitehead double of figure-8 http://upload.wikimedia.org/wikipedia/commons/thumb/a/ad/B_sat2.png/200px-B_sat2.png

and cabelling: cable of a connect-sum of trefoil and figure-8 http://upload.wikimedia.org/wikipedia/commons/thumb/5/5d/B_sat3.png/350px-B_sat3.png

There are many $n$-ary operations. The first one noticed (historically) is connect-sum: connect-sum of figure-8 and trefoil http://upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Sum_of_knots3.svg/300px-Sum_of_knots3.svg.png. The issue one might have with the $n$-ary connect-sum is its generated by the 2-ary connect sum. So here is an example of a $2$-ary splice that's not a connect sum: alt text http://upload.wikimedia.org/wikipedia/commons/thumb/a/a3/B_sat1.png/250px-B_sat1.png

There are a countable infinite collection of $n$-ary splices for any $n \geq 1$, moreover, there's still countably-infinite many primitive ones for any $n \geq 1$, where primitive means "can't be expressed in terms of $j$-ary operations for j less than n" These primitive splicing operations turn out to be specified (uniquely) by hyperbolic $(n+1)$-component links in the 3-sphere $L \subset S^3$, $L=L_0 \sqcup L_1 \sqcup \cdots \sqcup L_n$ such that the sublink $L_1 \sqcup \cdots \sqcup L_n$ is the trivial link. The hyperbolicity means $S^3 \setminus L$ has a complete hyperbolic structure of finite volume.

Splicing can be put in an operadic framework and this is the topic of one of my papers. So you can turn it into a purely algebraic formalism as well, by taking the homology of the space of all knots and the splicing operad, respectively.

It's not clear to me there's any reason for the seeming prevalence of 2-ary operations in mathematics. It appears to be more of an accident -- two things interacting is simpler, easier to contemplate.

In knot theory, splicing generally has more than just one or two inputs.

Splicing with one input generates things like Whitehead doubles:

and cabelling:

There are many $n$-ary operations. The first one noticed (historically) is connect-sum: . The issue one might have with the $n$-ary connect-sum is its generated by the 2-ary connect sum. So here is an example of a $2$-ary splice that's not a connect sum:

There are a countable infinite collection of $n$-ary splices for any $n \geq 1$, moreover, there's still countably-infinite many primitive ones for any $n \geq 1$, where primitive means "can't be expressed in terms of $j$-ary operations for j less than n" These primitive splicing operations turn out to be specified (uniquely) by hyperbolic $(n+1)$-component links in the 3-sphere $L \subset S^3$, $L=L_0 \sqcup L_1 \sqcup \cdots \sqcup L_n$ such that the sublink $L_1 \sqcup \cdots \sqcup L_n$ is the trivial link. The hyperbolicity means $S^3 \setminus L$ has a complete hyperbolic structure of finite volume.

Splicing can be put in an operadic framework and this is the topic of one of my papers. So you can turn it into a purely algebraic formalism as well, by taking the homology of the space of all knots and the splicing operad, respectively.

It's not clear to me there's any reason for the seeming prevalence of 2-ary operations in mathematics. It appears to be more of an accident -- two things interacting is simpler, easier to contemplate.