Singular instanton knot and sutured closure data

Assume full AC for the instanton constructions. Let $K\subset S^3$ be a smooth knot with a marked point $x$ and a preferred nonzero normal vector $v$ at $x$. Choose a small disk $D$ normal to $K$ at $x$, let $L=\mathrm{\partial }D$, and let $\omega$ be the radius of $D$ from $x$ to a point of $L$. Then $K^{\mathrm{♮}}:=K\cup L$ is the based earring link. Choose disjoint closed tubular neighborhoods $N(K)$ and $N(L)$ of its two components, set $X=S^3\setminus \mathrm{int}(N(K)\cup N(L))$, and truncate the endpoints of $\omega$ at $\mathrm{\partial }X$ to obtain $[\omega ]\in H_1(X,\mathrm{\partial }X;\mathbb{Z}\mathrm{/}2)$.

Here $PU(2)=U(2)\mathrm{/}U(1)\cong SO(3)$; a principal $PU(2)$ bundle $P\to X$ is equivalently an oriented real 3-plane bundle, and $w_2(P)\in H^2(X;\mathbb{Z}\mathrm{/}2)$ denotes its second Stiefel–Whitney class. Poincaré–Lefschetz duality for the oriented compact 3-manifold $X$ with boundary is the isomorphism $PD:H_1(X,\mathrm{\partial }X;\mathbb{Z}\mathrm{/}2)\to H^2(X;\mathbb{Z}\mathrm{/}2)$. Include a bundle $P$ with $w_2(P)=PD([\omega ])$ in the singular bundle data.

Such a bundle exists for every marked earring. Orient the truncated arc as a relative integral cycle and set $c=P{D}_{\mathbb{Z}}[\omega ]\in H^2(X;\mathbb{Z})$. Choose a complex line bundle $Q\to X$ with $c_1(Q)=c$, and let $P$ be the projective unitary frame bundle of $Q\oplus \underset{‾}{\mathbb{C}}$. On each 2-cell, a clutch of $Q$ with winding $n$ gives the projective transition whose $SU(2)$ lift ends at $(-1{)}^n$; its lifting obstruction is therefore $n<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2$. Hence $w_2(P)=c<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2=P{D}_{{\mathbb{F}}_2}[\omega ]$, as required.

In a local meridional angular coordinate $\theta$, an $SU(2)$ lift of the singular connection has the standard model $A_0+\frac{1}{4}\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right)d\theta$, where $A_0$ is smooth in an $O(2)$ reduction. As the radius of a meridional circle tends to zero, its holonomy conjugacy class tends to that of $\mathrm{diag}(i,-i)$ in $SU(2)$; the limiting image has order two in $PU(2)$. This is the prescribed asymptotic meridional holonomy, not an assertion of exact holonomy on every finite-radius circle for an arbitrary smooth $A_0$. For a flat connection the holonomies of homotopic meridional circles are conjugate, so their conjugacy class is exactly this limiting class. Define $I^{\mathrm{♮}}(K;\mathbb{Z})$ to be the singular instanton Floer homology $I^{\omega }(S^3,K^{\mathrm{♮}};P)$ associated to these data by Analytic foundations for the singular instanton knot complex, which supplies its analytic complex and independence of auxiliary choices under full AC. This does not assert sutured closure independence or a distinguished canonical class.

For the sutured version, let $E_K$ be the compact knot exterior and put two oppositely oriented meridional sutures $\gamma$ on its boundary torus. Choose a connected oriented annulus $T$, attach $[-1,1]\times T$ along the annular sutured region, and glue the two resulting boundary surfaces by an orientation-compatible diffeomorphism fixing a chosen point from $T$. The result is a closed oriented 3-manifold $\bar{Y}$ with a distinguished torus $\bar{R}$; the fixed point determines a standard circle meeting $\bar{R}$ once. Choose a line bundle $w\to \bar{Y}$ whose first Chern class is Poincaré dual to that circle, so $⟨c_1(w),[\bar{R}]⟩$ is odd. Let $I_{\ast }(\bar{Y},w;\mathbb{Q})$ denote the closed instanton Floer group with rational coefficients, using the admissible fixed-determinant theory in the external input below. For a homology class $z$ of dimension $0$ or $2$ in $\bar{Y}$, let $\mu (z)$ denote the standard instanton Floer operator associated to $z$; the point and surface operators commute and have degrees $4$ and $-2$, respectively. If $y\in \bar{R}$, define the closure group $KH{I}_{\bar{Y},\bar{R},w}(K;\mathbb{Q})$ by $KH{I}_{\bar{Y},\bar{R},w}(K;\mathbb{Q}):=\{a\in I_{\ast }(\bar{Y},w;\mathbb{Q}):\mu (\bar{R}{)}^{N}a=0\text{ for some }N\ge 1,\mu (y)a=2a\}\mathrm{.}$ This uses the generalized $0$-eigenspace of the surface operator and the ordinary $+2$-eigenspace of the point operator. For this admissible torus closure it equals the simultaneous ordinary $(0,2)$ eigenspace, by the extreme-surface simplicity input stated below; no global diagonalizability of either operator is assumed. The closure group is considered up to isomorphism, with independence of these auxiliary choices supplied by the external input below and the stated well-definedness justifier; no canonical vector or preferred basis is part of this definition.

Integral singular complex and continuation invariance ↗

Chosen compact knot exterior

Meridian and longitude conventions

Second Stiefel–Whitney class of the associated real 3-plane bundle

Integral and mod-two duals of the marked relative arc

Existence of a line bundle with the prescribed first Chern class

Hatcher, Algebraic Topology, Theorem 3.43

Kronheimer–Mrowka, Khovanov homology is an unknot-detector (2011), §§2.1–2.4, 4.2–4.4, and 8.7

Kronheimer–Mrowka, Knots, sutures and excision (2008), §§7.1–7.6

Kronheimer–Mrowka, Knots, three-manifolds and instantons (2018), §§4.3 and 5.1–5.2

Analytic foundations for the singular instanton knot complex ↗