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This question is about the beaviour of 4-genus of knots with respect to connected sum.

Let us indicate with $T(k)$ a Torus knot of type $(2,k)$, $k$ is an odd integer. Fix an orientation for every $T(k)$ so that $T(-k)$ represents the same knot with reversed orientation.

$T(k)\sharp T(-k)$ is a slice knot. More generally if $K^* $ denotes the mirror of $K$ then $K\sharp K^*$ is slice (infact ribbon).

My question is:

  • Is it true that a knot of the form $T(a_1)\sharp\dots\sharp T(a_n)$ is slice if and only if n is even, say $n=2k$, and we can arrange the coefficients so that for every $i\leq n$ we have $a_{k+i}=-a_i$?

Note that for any pair of knots $K_1$ and $K_2$, if both $K_1$ and $K_1\sharp K_2$ are slice then $K_2$ is also slice. Therefore one only needs to show that there exists $i$ and $j$ such that $a_i=-a_j$.

Of course we can generalize this problem:

  • Which connected sums of torus knots are slice?

Here are some links for definitions:

Torus knot:

Slice knot:

Slice genus:

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I imagine that's a hard problem. A fairly analogous problem of determining which connect-sums of lens spaces bound rational homology balls was completed in 2007 by Lisca. But being in Pisa I imagine you're well aware of that. – Ryan Budney Dec 8 '10 at 19:12
My first line of attack would be the Alexander polynomial. This is known for torus knots, and it is also known that a slice knot must have Alexander polynomial of the form $f(t)f(t^{-1})$. This should tell you quite a bit about what connect sums could be slice, although it won't be able to distinguish between a knot and its mirror image, so you'd need some other invariant to take care of this. – Jim Conant Dec 8 '10 at 19:19
Oh, this problem is easier than I thought. Litherland computed the group structure of the subgroup of the concordance group spanned by the torus knots, in 1979. See Kearton's survey, here: – Ryan Budney Dec 8 '10 at 19:33
@ Ryan, yes I am! – Paolo Aceto Dec 8 '10 at 20:58
@Jim, I have tried using the alexander polynomial, but I didn't go so far. – Paolo Aceto Dec 8 '10 at 21:09
up vote 10 down vote accepted

I believe the answer to your last two questions is yes, and it follows from Litherland's (1979) computation of the Tristram-Levine invariants of torus knots. See Kearton's survey here:

i.e. a connect sum of torus knots is slice if and only if the prime summands appear in balancing mirror-reflected pairs.

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Thanks for your answer! – Paolo Aceto Dec 9 '10 at 10:51

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