Narrow Results

Tangles were introduced by J. Conway. In 1970, he proved that every rational 2-tangle defines a rational number and two rational 2-tangles are isotopic if and only if they have the same rational number. So, from Conway's result we have a perfect classification for rational 2-tangles. However, there is no similar theorem to classify rational 3-tangles. In this paper, we introduce an invariant of rational n-tangles which is obtained from the Kauffman bracket. It forms a vector with Laurent polynomial entries. We prove that the invariant classifies the rational 2-tangles and the reduced alternating rational 3-tangles. We conjecture that it classifies the rational 3-tangles as well.

The Tait conjecture states that reduced alternating diagrams of links in $S^3$ have the minimal number of crossings. It has been proved in 1987 by M. Thistlethwaite, L. H. Kauffman and K. Murasugi studying the Jones polynomial. In this paper, we prove an analogous result for alternating links in $S^1\times S^2$ giving a complete answer to this problem. In $S^1\times S^2$ we find a dichotomy: the appropriate version of the statement is true for ${\mathbb{Z}}_2$-homologically trivial links, and our proof also uses the Jones polynomial. On the other hand, the statement is false for ${\mathbb{Z}}_2$-homologically non-trivial links, for which the Jones polynomial vanishes.

The Tait conjecture states that alternating reduced diagrams of links in $S^3$ have the minimal number of crossings. It has been proved in 1987 by Thistlethwaite, Kauffman and Murasugi studying the Jones polynomial. In [A. Carrega, The Tait conjecture in $S^1\times S^2$, J. Knot Theory Ramifications25(11) (2016) 1650063], the author proved an analogous result for alternating links in $S^1\times S^2$ giving a complete answer to this problem. In this paper, we extend the result to alternating links in the connected sum ${\#}_g(S^1\times S^2)$ of $g$ copies of $S^1\times S^2$. In $S^1\times S^2$ and ${\#}_2(S^1\times S^2)$, the appropriate version of the statement is true for ${\mathbb{Z}}_2$-homologically trivial links, and the proof also uses the Jones polynomial. Unfortunately, in the general case, the method provides just a partial result and we are not able to say if the appropriate statement is true.