Die gordische Auflösung von Knoten

Skein module deformations of elementary moves on links

“…Theorem 3.4 [29,33] For a knot K , its Kauffman skein quadruplet, K + , K − , K 0 , K ∞ and invariants u(K), ǫ(K) and λ(K) we have the following properties of the unknotting number: Proof Part (i) is a special case of the Wendt theorem, the first nontrivial result concerning unknotting number [40] (compare Lemma 2.2(h) of [25]) 13 . Assumptions of part (ii) guarantee that λ(K − ) = λ(K + ) − 1 as λ(K) can be changed at most by 1 when K is modified by a crossing change.…”

“…the link 9 2 40 in Rolfsen's notation [31]) are related by a (2, 2)-move; Figure 1.8. Similarly, the knot 9 40 and the closure of the 3-braid…”

The unknotting number and classical invariants, I

“…Let K be a knot. The first lower bounds on the unknotting number u(K) were given by Wendt [We37] who showed that u(K) ≥ minimal number of generators of H 1 (Σ(K); Z).…”

“…We will now quickly summarize all previous classical lower bounds on the unknotting number which are known to the authors. The first lower bounds on the unknotting number go back to Wendt [We37], they are subsumed by the following inequality due to Nakanishi [Na81]:…”

Graphs, determinants of knots and hyperbolic volume

“…This number is often called the determinant of L because of its expression (up to sign) as the determinant of a Seifert matrix [Rolfsen 1976, page 213] or a Goeritz matrix [Gordon and Litherland 1978]. The group H 1 (D L ) carries much interesting information on the link (in particular conditions for sliceness [Rolfsen 1976] and achirality [Hartley and Kawauchi 1979;Stoimenow 2005], and unknotting number estimates [Wendt 1937]). The determinant can be alternatively evaluated from the Jones polynomial V [1985] as |V (−1)|, and for alternating diagrams it has a nice combinatorial interpretation (see [Krebes 1999]) in terms of the Kauffman state model [1987].…”