Many links in space bound more than one minimal genus Seifert surface up to isotopy. A natural question is then to ask how similar the different minimal genus Seifert surfaces of a given link are. We will say that a surface S is annular of type k,l,m,… if S is obtained by Murasugi-summing A(k),A(l),A(m),… where A(n) is an unknotted annulus with n full twists. We will prove that if a link bounds two annular surfaces, then these surfaces must have the same type. This will be a corollary of a wider theorem: top coefficients of (certain) “non-semisimple” quantum link polynomials are Murasugi-sum multiplicative. This property marks a stark difference between this class of invariants and the (semisimple) quantum invariants of Jones and HOMFLY.
