Circular thin position for knots inS³

Additivity of Circular Width

“…Thus we have: Moreover, in [1] it is proved that if K 1 and K 2 are in circular thin decomposition, the circular handle decomposition induced on K 1 K 2 is circular locally thin. Thus, is natural to ask if such a decomposition is the thinnest for K 1 K 2 and if equality in (3.1) holds.…”

“…This concept was introduced by the second author in [1]. For sake of completeness we include some definitions and results.…”

“…In [1] the second author defined circular thin position and circular width for a knot in S 3 . The idea is to find collections of surfaces {S i } n i=1 and {F i } n i=1 , not necessarily connected, which are properly embedded in the knot exterior, such that each F i and each S i contains a Seifert surface for the knot.…”

“…In [1] an upper bound for the circular width of K 1 K 2 is given, which depends on the circular width of the original knot exteriors. Namely; cw(E(K 1 K 2 )) ≤ cw(E(K 1 )) cw(E(K 2 )) (1.1)…”

Counterexamples to Kauffman's conjectures on slice knots

“…Since R has a unique genus one Seifert surface, F and Σ are isotopic in the exterior of R. Then, by [30,Lemma 3.9], F and Σ cobound a product region P . At least one of the η i must lie in P because otherwise F would be isotopic to Σ in the exterior of the η i so F would be isotopic to F in the exterior of R , contradicting our assumption.…”