Framed chord diagrams and four-term relations

For a commutative ring $R$, a degree-$n$ chord diagram is an oriented circle with $n$ disjoint unordered pairs of distinct points, called chords, considered up to orientation-preserving homeomorphism of the circle. A chord records a double point by pairing its two preimages. The diagrams used for symbols of both framed and unframed invariants are these ordinary chord diagrams. In a framed singular-knot homotopy, each simple zero of the normal section may be marked by an auxiliary dot and counted as one singularity; these auxiliary dots are not generators of the chord-diagram modules below.

For $n\ge 2$, choose any degree-$(n-2)$ background chord diagram and choose three pairwise distinct new points $p,q,r$ on its circle, disjoint from all background endpoints. Choose disjoint endpoint-free neighborhoods of $p$ and $q$ that avoid $r$ and every background endpoint. Let $s_1,s_2,s_3,s_4$ lie, respectively, immediately before $p$, immediately after $p$, immediately before $q$, and immediately after $q$ in the circle orientation, each within the corresponding neighborhood. Let $D_i$ be the degree-$n$ diagram obtained by adding the fixed chord $pq$ and moving chord $r{s}_i$ to the background. The 4T relation is

$$D_1-D_2+D_3-D_4=0.$$

Two chords intersect if their endpoints alternate around the oriented circle. A chord is isolated if it intersects no other chord. The 1T relation sets every unframed diagram with an isolated chord equal to zero. The framed degree-$n$ chord-diagram module is the free $R$-module on degree-$n$ chord diagrams modulo all 4T relations. The unframed degree-$n$ chord-diagram module is the free $R$-module on degree-$n$ chord diagrams modulo 4T and 1T relations. A framed weight system is an $R$-linear functional on the framed module; an unframed weight system is an $R$-linear functional on the unframed module.

The figure shows one generator of 4T. The solid chord is fixed; the dashed chord has fixed endpoint $r$ and moving endpoint $s_i$. The arrow gives the positive orientation of the circle. The same background chords may be inserted identically in every panel.

Singular-knot, framing-zero, and resolution conventions

Bar-Natan and Stoimenow, §1.1, pp. 3–5

Chmutov, Duzhin, and Mostovoy, §§3.4, 3.5.1, and 4.1, pp. 70–75 and 89–91