When is a connected sum of torus knots a slice knot? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T22:03:47Z http://mathoverflow.net/feeds/question/48664 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48664/when-is-a-connected-sum-of-torus-knots-a-slice-knot When is a connected sum of torus knots a slice knot? Paolo Aceto 2010-12-08T18:59:31Z 2010-12-09T10:42:21Z <p>This question is about the beaviour of 4-genus of knots with respect to connected sum.</p> <p>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.</p> <p>$T(k)\sharp T(-k)$ is a slice knot. More generally if $K^*$ denotes the mirror of $K$ then $K\sharp K^*$ is slice (infact ribbon).</p> <p>My question is:</p> <ul> <li>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 $a_{k+i}=-a_i$? </li> </ul> <p>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 $a_i=-a_j$.</p> <p>Of course we can generalize this problem:</p> <ul> <li>Which connected sums of torus knots are slice?</li> </ul> <p>Here are some links for definitions:</p> <p>Torus knot: <a href="http://en.wikipedia.org/wiki/Torus_knot" rel="nofollow">http://en.wikipedia.org/wiki/Torus_knot</a></p> <p>Slice knot: <a href="http://en.wikipedia.org/wiki/Slice_knot" rel="nofollow">http://en.wikipedia.org/wiki/Slice_knot</a></p> <p>Slice genus: <a href="http://en.wikipedia.org/wiki/Slice_genus" rel="nofollow">http://en.wikipedia.org/wiki/Slice_genus</a></p> http://mathoverflow.net/questions/48664/when-is-a-connected-sum-of-torus-knots-a-slice-knot/48667#48667 Answer by Ryan Budney for When is a connected sum of torus knots a slice knot? Ryan Budney 2010-12-08T19:34:29Z 2010-12-08T19:40:03Z <p>I believe the answer to your last two questions is <em>yes</em>, and it follows from Litherland's (1979) computation of the Tristram-Levine invariants of torus knots. See Kearton's survey here: </p> <p><a href="http://www.maths.ed.ac.uk/~aar/slides/durham.pdf" rel="nofollow">http://www.maths.ed.ac.uk/~aar/slides/durham.pdf</a></p> <p>i.e. a connect sum of torus knots is slice if and only if the prime summands appear in balancing mirror-reflected pairs. </p>