PacificJournal ofMathematics

A NOTE ON MURASUGI SUMS

ABIGAIL A. THOMPSON

Volume 163 No. 2 April 1994

PACIFIC JOURNAL OF MATHEMATICSVol. 163, No. 2, 1994

A NOTE ON MURASUGI SUMS

ABIGAIL T H O M P S O N

We give two examples to show that the genus of knots is neithersub- nor super-additive under the Murasugi sum operation.

A number of "addition" operations can be defined on pairs of knotsin S3 the connected sum is the most obvious of these, but there areseveral other more complicated possibilities. A general question onecan ask is: which properties of knots behave "nicely" under these op-erations? It has long been known that the genus of a knot is additiveunder connect sum. Schubert [Sc] showed that bridge number is ad-ditive minus one under connect sum.

Outstanding questions are how crossing number, unknotting num-ber and tunnel number behave under connect sum. Only the mostobvious inequalities are currently available, and they are quite weak—for example, the crossing number is obviously sub-additive, as is theunknotting number, and it is easy to show that the tunnel number ofthe connect sum of K\ and K>ι is less than or equal to the sum oftheir tunnel numbers plus one.

A more complicated operation on pairs of knots is the band-connectsum. This operation is not well defined, since it depends on how theband is chosen. Gabai and Scharlemann simultaneously establishedthe superadditivity of genus under band-connect sum [Gl], [S].

Yet another operation combining knots is the Murasugi sum of twoknots (see [G2] for a definition); this depends on a choice of Seifertsurfaces for the knots as well as a choice of disks along which to do thesum. Gabai [G2] nevertheless has shown that under reasonable con-ditions many geometric properties of the Seifert surfaces are retainedunder the Murasugi sum. In particular, he has shown that the Mura-sugi sum of K\ and Kι along minimal genus Seifert surfaces R\ andi?2 yields a minimal genus Seifert surface R for the resulting knotK, so genus is additive under Murasugi sum provided the addition isdone along minimal genus surfaces. Taking the Murasugi sum of twoknots can thus be considered a "natural" operation on pairs consistingof knots together with minimal genus Seifert surfaces. However, the

393

394 ABIGAIL THOMPSON

FIGURE 1

OFIGURE 2

operation of constructing a Murasugi sum is not confined to minimalgenus or even incompressible Seifert surfaces; we give two examplesto illustrate that the genus does not behave in a predictable way in thislarger category. The first [Figure 1] is an example of two trivial knots,

MURASUGI SUMS 395

each bounding a (compressible) genus one surface, summed along asquare to yield a trefoil. The second example [Figure 2] is two figureeight knots, one bounding a genus one surface and the other boundinga (compressible) genus two surface, summed along a square to yieldthe trivial knot.

REFERENCES

[Gl] D. Gabai, Genus is superadditive under band connected sum, Topology, 26,No. 2 (1987), 209-210.

[G2] , The Murasugi sum is a natural geometric operation, Contemp. Math.,vol. 20, Amer. Math. Soc, Providence, RI, 1983, pp. 131-143.

[S] M. Scharlemann, Sutured manifolds and generalized Thurston norms, J. Dif-ferential Geom., 29 (1989), 557-614.

[Sc] H. Schubert, Uber eine numerische Knoteninvariante, Math. Z., 61 (1954),245-288.

Received April 4, 1992. Partially supported by an NSF grant and a grant from theAlfred P. Sloan Foundation.

UNIVERSITY OF CALIFORNIADAVIS, CA 95616

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Vol. 163, No. 2 April 1994

PACIFIC JOURNAL OF MATHEMATICS

Volume 163 No. 2 April 1994

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237On sieved orthogonal polynomials. X. General blocks of recurrencerelations

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393A note on Murasugi sumsABIGAIL A. THOMPSON

0030-8730(1994)163:2;1-I

PacificJournalofM

athematics

1994Vol.163,N

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