Forks, noodles and the LKB intersection pairing

Let $D$, $P=\{p_1,\dots ,p_n\}$ and $C$ be as in The two-point configuration space of a punctured disk, with the chosen boundary points $d_1,d_2$ of the lower arc, and let $\tilde{C}\to C$ be the LKB cover with its pairing modules $H_2(\tilde{C},\tilde{\nu })$ and $H_2(\tilde{C},\mathrm{\partial }\tilde{C}\cup \tilde{\nu })$ of The relative pairing modules as stabilized direct limits.

Noodles. A noodle is an embedded edge $N\subset D$ with endpoints $d_1,d_2$, whose interior lies in $D\setminus P$. Every noodle is oriented from $d_1$ to $d_2$. Its surface is $\mathrm{\Sigma }(N)=\{\{x,y\}\in C:x,y\in N,x\mathrm{\ne }y\}\subset C,$ oriented by the orientation of $N$ as in Bigelow 2001 section 2, and $\tilde{\mathrm{\Sigma }}(N)$ denotes the lift of $\mathrm{\Sigma }(N)$ containing ${\tilde{c}}_0$. The proper triangle has a collision end along its omitted diagonal. Its end-stable class is therefore $y_N=[\tilde{\mathrm{\Sigma }}(N)]\in H_2(\tilde{C},\mathrm{\partial }\tilde{C}\cup \tilde{\nu }),$ not ordinary boundary-only homology. Parametrize $N$ by $I=[0,1]$ and truncate to $0\le u<v\le 1$, $v-u\ge \delta$. This compact triangle lifts from $c_0$; its outer sides lie in $\mathrm{\partial }\tilde{C}$ and its third side lies in ${\tilde{\nu }}_{\epsilon }$ when $\delta$ is sufficiently small by uniform continuity. Differences of smaller truncations lie in that collision end, so the finite relative cycles define the compatible stabilized class.

Forks. A fork is an embedded tree $F\subset D$ with four vertices $d_1,p_i,p_j,z$, such that $F\cap \mathrm{\partial }D=\{d_1\}$, $F\cap P=\{p_i,p_j\}$, and all three edges of $F$ have $z$ as a vertex. The edge containing $d_1$ is the handle of $F$; the union of the other two edges is the tine edge $T(F)$, an embedded edge from $p_i$ to $p_j$ through $z$. The tine edge is oriented so that the handle lies to its right. A parallel copy of $F$ is a parallel tree $F^{\mathrm{\prime }}$ whose handle starts at $d_2$, obtained by pushing the tine and handle of $F$ off themselves and then translating the two tine endpoints through $P$ along the respective tine ends, as in Bigelow 2001 Figure 1; write $z^{\mathrm{\prime }}$ for its trivalent vertex and $T(F^{\mathrm{\prime }})$ for its tine edge. The surface of the fork is $\mathrm{\Sigma }(F)=\{\{x,y\}\in C:x\in T(F)\setminus P,y\in T(F^{\mathrm{\prime }})\setminus P\}\subset C,$ homeomorphic to the interior of the square $T(F)\times T(F^{\mathrm{\prime }})$ and oriented by the two tine orientations. Let ${\beta }_1$ be the arc from $d_1$ to $z$ along the handle of $F$ and ${\beta }_2$ the arc from $d_2$ to $z^{\mathrm{\prime }}$ along the handle of $F^{\mathrm{\prime }}$, and let $\tilde{\beta }$ be the lift of the arc $\{{\beta }_1,{\beta }_2\}$ in $C$ starting at ${\tilde{c}}_0$; the lifted surface $\tilde{\mathrm{\Sigma }}(F)$ is the lift of $\mathrm{\Sigma }(F)$ containing $\tilde{\beta }(1)$. Thus a fork presents a class $[\tilde{\mathrm{\Sigma }}(F)]\in H_2(\tilde{C},\tilde{\nu })$; the closed compact replacement of a multiple of this class is the subject of A multiple of a fork surface has a closed compact replacement.

The LKB pairing. For $x\in H_2(\tilde{C};\mathbb{Z})$ and $y\in H_2(\tilde{C},\mathrm{\partial }\tilde{C}\cup \tilde{\nu })$ let $x\cdot y\in \mathbb{Z}$ denote the algebraic intersection number of representatives in general position, and let $q^a{t}^{b}y$ denote the image of $y$ under the deck transformation $q^a{t}^b$. The LKB pairing is $⟨x,y⟩={\sum }_{a,b\in \mathbb{Z}}(x\cdot q^a{t}^{b}y)q^a{t}^b\in \mathrm{\Lambda }\mathrm{.}$ Interchanging the two variables with the two relative modules gives the primed pairing $⟨\cdot ,\cdot {⟩}^{\mathrm{\prime }}:H_2(\tilde{C},\tilde{\nu })\times H_2(\tilde{C},\mathrm{\partial }\tilde{C})⟶\mathrm{\Lambda },⟨x^{\mathrm{\prime }},y^{\mathrm{\prime }}{⟩}^{\mathrm{\prime }}={\sum }_{a,b\in \mathbb{Z}}(x^{\mathrm{\prime }}\cdot q^a{t}^b{y}^{\mathrm{\prime }})q^a{t}^b\mathrm{.}$ The geometric fork/noodle polynomial in finite transverse position is the signed sum of its labelled deck intersections. For a closed absolute replacement $c_F$ of ${\mathrm{\Delta }}_F\tilde{\mathrm{\Sigma }}(F)$, with ${\mathrm{\Delta }}_F=(1-q{)}^2(1+qt)$, it satisfies $⟨c_F,y_N⟩={\mathrm{\Delta }}_F⟨N,F⟩\mathrm{.}$ This uses the absolute/end-stable pairing, not a generic pairing of two end-relative modules. The finite diagram sum and this identity are verified in The fork-noodle pairing is well defined and equivariant; the sum displayed here is shown to be finite, independent of representatives and $\mathrm{\Lambda }$-sesquilinear in that item.

Sesquilinearity. For $x,y,x^{\mathrm{\prime }},y^{\mathrm{\prime }}$ in the appropriate modules and $\lambda \in \mathrm{\Lambda }$ one has $⟨\lambda x,y⟩=\lambda ⟨x,y⟩=⟨x,\bar{\lambda }y⟩,⟨\lambda x^{\mathrm{\prime }},y^{\mathrm{\prime }}{⟩}^{\mathrm{\prime }}=\lambda ⟨x^{\mathrm{\prime }},y^{\mathrm{\prime }}{⟩}^{\mathrm{\prime }}=⟨x^{\mathrm{\prime }},\bar{\lambda }{y}^{\mathrm{\prime }}{⟩}^{\mathrm{\prime }},$ where $\bar{\lambda }(q,t)=\lambda (q^{-1},t^{-1})$; the second identity uses that the intersection number is additive in each variable and that the deck action on the second variable conjugates the coefficient. These identities are verified in The fork-noodle pairing is well defined and equivariant.

The relative modules occur in this page only as targets of the two pairings; the representation itself lives on the absolute module The integral LKB module as absolute second homology.