On diagram formulas for knot invariants

Explicit formulas for Vasil'ev invariants of the fourth order for knots

“…To prove this, it suffices to show the independence of the right-hand side under the Reidemeister motions. Let us verify the invariance of the arrow polynomial P 1 4 with respect to generating Reidemeister motions on the Gaussian diagram.…”

“…The first Reidemeister motion is the addition (annihilation) of a loop in the plane knot diagram or is the addition (annihilation) of an isolated arrow in the Gaussian diagrams, which does not influence on the number and signs of subdiagrams which make a contribution to the formula (in the expression for the polynomial P 1 4 , there are no arrow diagrams with isolated arrow) and does not change the value of V 1 4 . The second Reidemeister motion on the Gaussian diagram corresponds to the addition (annihilation) of two commonly directed arrows (intersecting or disjoint) with different signs between the ends and origins of which other arrows do not begin or finish.…”

“…Viro and M. B. Polyak obtained explicit diagramarrow formulas for invariants of the second and third order [6]. Still now, there are no diagram-arrow formulas for invariants of knots of the fourth order (see [1,2]), although attempts to obtain them were repeatedly undertaken and announced [6,7]. In [4], the author presents explicit combinatorial formulas of another type for invariants of the fourth order, but these formulas are still not proved.…”

“…The goal of the present paper is the proof of explicit diagram-arrow formulas for two integer-valued basic Vasil'ev invariants V 1 4 (K) and V 2 4 (K) of the fourth order for oriented knots. The coefficients in these formulas are obtained by analyzing overdetermined systems of linear equations with a large number of unknowns.…”

“…The basis of the vector space of invariants of the fourth order is equal to 3. Our formula for V 1 4 (K) has rational coefficients, although it defines an integer-valued invariant, which follows from its normalization. The fact that the invariant V 2 4 (K) is integer-valued directly follows from the formula.…”