Topological invariants of knots and links

Covering Sato-Levine invariants

“…However, the most striking property of the covering Sato-Levine invariants is that while the δ-function is independent of one's choice of the orientation of a given link, the τ -function is extremely sensitive to this and can be used to detect both nonamphicheirality and noninvertibility of two-component links even up to link concordance. This observation makes our τ -function quite different from most of the commonly used link invariants, including the 2-variable Alexander polynomial [1], the Milnor invariants [23] and the various quantum group invariants constructed after Jones's discovery of his famous Jones polynomial [14]; all these invariants fail to distinguish two-component links from their inverses.…”

Factorization of differential expansion for antiparallel double-braid knots

“…Modern group theory is incapable to provide the answers beyond pure symmetric and antisymmetric representations [97][98][99][100] [103] and [104] for some inclusive Racah matrices for R = [3,1] and R = [2, 2] respectively). Further progress on these lines seems to be beyond the current computer capacities.…”

“…with A-independent factor F (m,n) [1] . Thus Alexander polynomial Al (m,n) [1] = 1 + mn{q} 2 has degree one, and defect [110] of the differential expansion is zero for the entire family.…”

“…Construction of knot polynomials [1][2][3][4][5][6][7][8] is currently at the front-line of modern theoretical physics, because this is an exactly solvable problem in quantum field theory, which remains unsolved for years, despite tremendous effort by many distinguished researchers. Far ago it was reformulated in terms of Chern-Simons theory [9]- [17], where (at least for the knots in S 3 ) it is basically reduced to a free-field calculation, and a formal answer is provided [18]- [32] in terms of representation theory of quantum groups.…”

Heegaard Diagrams and Holomorphic Disks

“…Note that the function F (x) is the T -power appearing for the contribution of x to the symmetrized Alexander polynomial, see [2], [44]. The Maslov grading gr(x) is defined analogously.…”