4·2. Generalised index polynomials

We now define index polynomial invariants for generalised knotoids. Let ${\mathcal{D}}$ be an oriented, constituent-labelled generalised knotoid diagram. Let c be a crossing, and let $o_c$ and $u_c$ denote the constituents of ${\mathcal{D}}$ containing the over-strand and under-strand at c, respectively. Suppose an oriented smoothing is applied to c. Let $\alpha_c$ denote the constituent of the smoothed diagram containing the incoming portion of $o_c$ , and let $\beta_c$ denote the constituent of the smoothed diagram containing the incoming portion of $u_c$ . Note that $o_c$ and $u_c$ are not necessarily distinct, and $\alpha_c$ and $\beta_c$ are not necessarily distinct.

We introduce variables $t_{e}$ , and $s_{e}$ for constituents e and $r_{e_1, e_2}$ for pairs of (not necessarily distinct) constituents $(e_1, e_2)$ and define

\[g(c) = r_{o_c, u_c}\left(\prod_{e \in C({\mathcal{D}})} t_e^{\alpha_c \cdot e} - 1\right)\left(\prod_{e \in C({\mathcal{D}})}s_e^{\beta_c \cdot e} - 1\right).\]

We define the generalised index polynomial by

(4·2·1) \begin{align}G_{\mathcal{D}}(r, s, t) = \sum_c \mathrm{sgn}(c)g(c).\end{align}

Here r represents a vector whose components are the variables $r_{e_1, e_2}$ in some fixed order, and s and t are similar. Operations on r, s, and t are applied component-wise. For example, the polynomial $G_{\mathcal{D}}(r, s^{-1}, t)$ is obtained by replacing $s_e$ with $s_e^{-1}$ for all $e \in C({\mathcal{D}})$ , and $G_{\mathcal{D}}(r, t, s)$ is obtained by swapping $t_e$ and $s_e$ for all $e \in C({\mathcal{D}})$ . For r, we let $\overline r$ indicate the operation that swaps $r_{e_1, e_2}$ and $r_{e_2, e_1}$ for all constituents $e_1, e_2$ .

The generalised index polynomial extends the affine index polynomial in the following sense.

Fig. 12. Two possible configurations for a Type II Reidemeister move. In each of Figures 12(a) and 12(b), the two diagrams obtained by smoothing at $c_1$ and $c_2$ carry the same algebraic intersection data.

Proof. If ${\mathcal{D}}$ is a spherical or planar knotoid diagram, the generalised index polynomial becomes a three-variable polynomial

\[G_{\mathcal{D}}(r,s,t) = \sum_c \mathrm{sgn}(c)r(t^{\alpha_c \cdot e} - 1)(s^{\beta_c \cdot e} -1 ).\]

Here, e is the single segment constituent of ${\mathcal{D}}$ . Because self-crossings of $\alpha_c$ create two intersections of $\alpha_c$ with e which cancel each other out, all contributions to $\alpha_c \cdot e$ come about through intersections of $\alpha_c$ with $\beta_c$ . Hence, $\alpha_c \cdot e = \alpha_c \cdot \beta_c = -\beta_c \cdot e = w(c)$ . It follows that the affine index polynomial $F^{\mathrm{aff}}_{\mathcal{D}}(t)$ is recovered from $G_{\mathcal{D}}(s,t)$ by

\[F^{\mathrm{aff}}_{\mathcal{D}}(t) = -G_{\mathcal{D}}(1, 0, t).\]

Conversely, $G_{\mathcal{D}}(r,s,t)$ is recovered from $F^{\mathrm{aff}}_{\mathcal{D}}(t)$ by

\[G_{\mathcal{D}}(r,s,t) = r(F^{\mathrm{aff}}_{\mathcal{D}}(ts^{-1}) - F^{\mathrm{aff}}_{\mathcal{D}}(t) - F^{\mathrm{aff}}_{\mathcal{D}}(s^{-1})).\]

By Proposition 4·1, the generalised index polynomial is trivial for any classical link.

If a generalised knotoid diagram ${\mathcal{D}}$ is the disjoint union of diagrams ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ with no crossings occuring between constituents of ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ , then the generalised index polynomial $G_{\mathcal{D}}$ provides no information on their relative positioning. For example, a valency-zero pole can be moved to any region of a diagram without changing the generalised index polynomial.

We extend the generalised index polynomial for pole-labelled generalised knotoids with at least one pole. Given a diagram ${\mathcal{D}}$ and a fixed pole $P \in P({\mathcal{D}})$ , we choose oriented shortcuts $\{\gamma_Q \mid Q \in P({\mathcal{D}})\}$ from P to each pole in ${\mathcal{D}}$ . For a constituent e, let $\textbf{1}_L(e)$ denote the indicator function that is 1 if e is a loop constituent and 0 otherwise. We introduce variables $a_Q, b_Q$ for each $Q \in P({\mathcal{D}})$ and define, for each crossing c, the polynomial

\[h_P(c) = \left(\prod_{Q \in P({\mathcal{D}})} a_Q^{\gamma_Q \cdot \alpha_c}\right)^{\textbf{1}_L(\alpha_c)}\left(\prod_{Q \in P({\mathcal{D}})} b_Q^{\gamma_Q \cdot \beta_c}\right)^{\textbf{1}_L(\beta_c)}.\]

We define the base-pointed index polynomial by

(4·2·2) \begin{equation} G_{{\mathcal{D}}, P}(r,s,t,a,b) = \sum_c \mathrm{sgn}(c)g(c)h_P(c).\end{equation}

The notational remarks following Equation (4·2·1) apply to the variables a and b as well.

By Corollary 4·2, the base-pointed index polynomial is independent of the choice of shortcuts $\gamma_Q$ , and the proof that $G_{{\mathcal{D}}, P}$ is a generalised knotoid invariant is identical to the proof of Theorem 4·3. The choice of the base-point pole P is immaterial. If P ^′ is another pole, the polynomial $G_{{\mathcal{D}}, P'}$ is obtained from $G_{{\mathcal{D}}, P}$ in the following way. Since we can take as shortcut from P ^′ to any other pole Q the path obtained by reversing the orientation of $\gamma_{P'}$ and then adding to it the shortcut from P to Q, this modifies each monomial in $G_{{\mathcal{D}}, P}$ by subtracting the exponent of $a_{P'}$ from the exponent of each $a_Q$ and subtracting the exponent of $b_{P'}$ from the exponent of each $b_Q$ .

Example 4·5. The generalised knotoid $\kappa$ in Figure 13(a) has

\[G_{\kappa, P} = r_{e_1, e_1}((s_{e_2}^2 - 1)a_Q^{-1}a_R^{-1}b_Qb_R^{-1} - (t_{e_2} - 1)(s_{e_2}-1)a_Q^{-1}a_R^{-2}b_Q)\]

and

\[G_{\kappa, R} = r_{e_1, e_1}((s_{e_2}^2 - 1)a_Pb_Pb_Q^2 - (t_{e_2} - 1)(s_{e_2}-1)a_P^2a_Q^1b_Q).\]

Fig. 13. Finding polynomials of certain generalised knotoids.

We now define a variant of the base-pointed index polynomial. With all notation as before, define

\[\tilde g(c) = r_{o_c, u_c}\prod\limits_{e \in C({\mathcal{D}})}t_e^{\alpha_c \cdot e}s_e^{\beta_c \cdot e}\]

and

\[\tilde h_P(c) = \left(\textbf{1}_L(\alpha_c)\prod_{Q \in P({\mathcal{D}})} a_Q^{\gamma_Q \cdot \alpha_c} - 1\right)\left(\textbf{1}_L(\beta_c)\prod_{Q \in P({\mathcal{D}})} b_Q^{\gamma_Q \cdot \beta_c} -1\right).\]

We define the pole-centric base-pointed index polynomial by

\[\tilde G_{{\mathcal{D}}, P}(r,s,t,a,b) = \sum_c \mathrm{sgn}(c)\tilde g(c) \tilde h_P(c).\]

The proof that $\tilde G_{{\mathcal{D}}, P}$ is a generalised knotoid invariant is nearly identical to the proof of Theorem 4·3. The remarks following Equation (4·2·2) for $G_{{\mathcal{D}}, P}$ hold for $\tilde G_{{\mathcal{D}}, P}$ as well.

In analogy to Proposition 4·4, the pole-centric base-pointed index polynomial extends the strengthened index polynomial $F^{\mathrm{ind}}(s,t)$ for knotoids. Suppose $k_1$ and $k_2$ are two spherical knotoids, each with poles labelled L and H and a constituent oriented from L to H. In what follows, we take $\alpha_c$ and $\beta_c$ to be as in the definition of the generalised index polynomial as in Section 4·2, and we take ind(c) to be as defined in Kim, Im, and Lee’s index polynomial in point (iii) under Equation 4·1·1.

Proof. For a knotoid diagram ${\mathcal{D}}$ oriented from L to H, the pole-centric base-pointed index polynomial based at L is a five-variable polynomial $\tilde G_{{\mathcal{D}}, L}(r, s, t, a_H, b_H)$ . Set $r=s=t = 1$ to obtain a two-variable polynomial $\tilde G_{{\mathcal{D}}, L}(a_H, b_H)$ . For each crossing c in ${\mathcal{D}}$ , exactly one of $\alpha_c, \beta_c$ is a loop constituent. The crossings for which $\alpha_c$ is the loop constituent is precisely the set of early undercrossings, and the crossings for which $\beta_c$ is the loop constituent is precisely the set of early overcrossings. Thus

\[\tilde G_{{\mathcal{D}}, L}(a_H, b_H) = -\sum_{c \in U(k)} \mathrm{sgn}(c)(a_H^{\gamma_{H} \cdot \alpha_c} - 1) - \sum_{c \in O({\mathcal{D}})} \mathrm{sgn}(c)(b_H^{\gamma_{H} \cdot e} - 1).\]

Suppose $\alpha_c$ is the loop constituent. Observe that $\beta_c$ defines an oriented shortcut from L to H. It follows from Corollary 4·2 that

\[\gamma_H \cdot \alpha_c = \beta_c \cdot \alpha_c = -\mathrm{ind(c)}.\]

Similarly, if $\beta_c$ is the loop constituent, then $\gamma_H \cdot \beta_c = \alpha_c \cdot \beta_c = -\mathrm{ind}(c)$ . Thus we obtain

\begin{align*} -\tilde G_{{\mathcal{D}}, L}(s^{-1}, t^{-1}) &= \sum_{c \in U({\mathcal{D}})} \mathrm{sgn}(c)(s^{\mathrm{ind}(c)} - 1) + \sum_{c \in O({\mathcal{D}})}\mathrm{sgn}(c)(t^{\mathrm{ind}(c)} - 1)\\ &= F^{\mathrm{ind}}_{\mathcal{D}}(s,t).\end{align*}

In particular, the strengthened index polynomial is recoverable from the pole-centric base-pointed index polynomial, and the conclusion follows.

The index polynomials of the form given by Equation (4·1·1) cannot distinguish between inequivalent planar knotoids with diagrams that are equivalent when considered on the sphere. By viewing a planar knotoid as a generalised knotoid with three poles L, H, and $\infty$ and a constituent oriented from L to H, as in Example 2·3, the base-pointed index polynomial and its pole-centric variant yield new invariants for planar knotoids. For example, the planar knotoid k in Figure 13(d) has trivial index polynomial but has $\tilde G_{k, \infty} = a_La_H$ .

Following the notation of [ Reference Turaev31 ] and [ Reference Linov27 ], we extend the basic knotoid involutions to generalised knotoids and state their effects on the generalised index polynomials. Given a diagram ${\mathcal{D}}$ , we let $\mathrm{mir}({\mathcal{D}})$ denote the diagram obtained by toggling all over/under crossing data, and we let $\mathrm{sym}({\mathcal{D}})$ denote the diagram obtained by reflecting across a great circle of $S^2$ (and preserving crossing data). We let rot denote the composition $\mathrm{mir} \circ \mathrm{sym} = \mathrm{sym} \circ \mathrm{mir}$ . For an oriented diagram ${\mathcal{D}}$ , we let $\mathrm{rev}({\mathcal{D}})$ denote the diagram obtained by reversing the orientation on all constituents. It is clear that mir, sym, rot, and rev define involutions on the set of generalised knotoids, possibly labelled or oriented. Note that mir, sym, and rot preserve the underlying directed graph of an oriented generalised knotoid, while rev may not.

Thus the base-pointed index polynomials may be used to distinguish between $\kappa$ and its images under these three basic involutions. But no such relationship holds for rev.

The base-pointed index polynomials also yield lower bounds on the height between poles in a generalised knotoid. Given a Laurent polynomial $F \in \mathbb Z[x_1^{\pm 1}, \dots, x_n^{\pm 1}]$ , we let $[x_i]F \in \mathbb Z[x_1^{\pm 1}, \dots, x_{i-1}^{\pm 1}, x_{i+1}^{\pm 1}, \dots, x_n^{\pm 1}]$ denote the coefficient of $x_i$ in F. For any subset $S = \{x_{i_1}, \dots, x_{i_m}\}$ of the variables and a monomial $M = \prod_{j=1}^n x_j^{u_j}$ , we define $\deg_S(M)=\sum_{k=1}^m |u_{i_k}|$ , and we let $\deg_S(F)$ denote the maximum of $\deg_S(M)$ over all monomials M with nonzero coefficient F. (We omit the set notation from S for brevity.)

Proposition 4·11. Let $\kappa$ be a pole-labelled, constituent-labelled, oriented generalised knotoid and let $P, Q \in P(\kappa)$ be poles. Then

\[h_\kappa(P,Q) \ge \sum_{e \in C(\kappa)} \deg_{a_Q, b_Q}([r_{e, e}]G_{\kappa, P}).\]

An identical inequality holds with $G_{\kappa, P}$ replaced by $\tilde G_{\kappa, P}$ .

Proof. The following argument holds for both $G_{\kappa, P}$ and $\tilde G_{\kappa, P}$ . Let ${\mathcal{D}}$ be any diagram representing $\kappa$ and let $\gamma$ be any shortcut oriented from P to Q. It suffices to show that, for any constituent $e \in C({\mathcal{D}})$ , the number of intersections $\#(\gamma \cap e)$ of $\gamma$ with e away from poles is at least $d \;:\!=\; \deg_{a_Q, b_Q}([r_{e,e}]G_{\kappa, P})$ .

If $d = 0$ , this is trivial. Assuming $d > 0$ , the diagram ${\mathcal{D}}$ must contain a crossing c with $o_c = u_c = e$ and $\textbf{1}_L(\alpha_c)|\gamma_Q \cdot \alpha_c| + \textbf{1}_L(\beta_c)|\gamma_Q \cdot \beta_c| = d$ . Note that $\alpha_c \neq \beta_c$ , and

\begin{align*} \#(\gamma \cap e) &= \#(\gamma \cap \alpha_c) + \#(\gamma \cap \beta_c)\\ &\ge |\gamma_Q \cdot \alpha_c| + |\gamma_Q \cdot \beta_c|\\ &\ge d. \end{align*}

Proposition 4·11 implies that the generalised knotoid $\kappa_1$ from Example 4·5 has $h_{\kappa_1}(P, Q) = h_{\kappa_1}(P, R) = h_{\kappa_1}(R,Q) = 2$ , and the generalised knotoid $\kappa_2$ from Example 13(c) has $h_{\kappa_2}(P, Q) = 2$ and $h_{\kappa_2}(P, R) = h_{\kappa_2}(Q,R) = 1$ .

For certain generalised knotoids $\kappa$ , the polynomials $G_{\kappa, P}$ or $\tilde G_{\kappa, P}$ give more information than the bound stated in Proposition 4·11, as the bound only considers intersections of a shortcut $\gamma$ with loops $\alpha_c, \beta_c$ that result from smoothing at a self-intersection of a constituent e. The generalised knotoid $\kappa$ in Figure 13(e) has $\tilde G_{\kappa, P} = 2r_{e_1, e_2}a_Q^{-2}$ , and an argument similar to the proof of Proposition 4·11 shows that $h_\kappa(P,Q) = 2$ despite the fact that Proposition 4·11 only yields the trivial bound $h_\kappa(P, Q) \ge 0$ .

We remark on non-constituent-labelled generalised knotoids. If $\kappa$ is simply an oriented, generalised knotoid, we obtain a two-variable generalised index polynomial $G_{\kappa}(s,t)$ by computing the generalised index polynomial of $\kappa$ with constituents arbitrarily labelled, and then setting $r_{e_1, e_2}=1$ , $t_{e_1}=t$ , and $s_{e_1} = s$ for all $e_1, e_2 \in C(\kappa)$ . In the case that $\kappa$ is pole-labelled, we again label the constituents arbitrarily and compute the base-pointed index polynomial (or the pole-centric variant). We then re-index the variables $r_{e_1, e_2}, t_e, s_e$ as follows. Each subscript e corresponding to a segment constituent e oriented from pole P to pole Q is replaced by (P, Q), and all subscripts corresponding to loop constituents are replaced by a single symbol L.

5·1. Generalised bracket polynomial

We define an extension of the Kauffman bracket polynomial for oriented, pole-labelled, edge-labelled generalised knotoids.

Fig. 14. A crossing, its A-type smoothing, and its B-type smoothing, respectively.

Given a generalised knotoid diagram ${\mathcal{D}}$ and a crossing c in ${\mathcal{D}}$ , a smoothing at c is one of the two operations pictured in Figure 14, and is either A-type or B-type. Observe that a smoothing preserves the number of segment constituents. A state s of ${\mathcal{D}}$ is a diagram obtained by smoothing at each of the crossings of ${\mathcal{D}}$ . Let $\sigma(s)$ denote the number of A-type smoothings of s minus the number of B-type smoothings of s. Observe that s has no crossings and has the same number of segment constituents as D, though in general, there is no natural correspondence between the segment constituents of D and s.

For every state s, we define auxiliary polynomials $G_s$ , $L_s$ , and $E_s$ to record the data of the state’s underlying graph, loop constituents, and segment constituents, respectively. They are as follows.

Introduce variables $\lambda_{\{P,Q\}}$ for every unordered pair $\{P, Q\}$ of not necessarily distinct poles $P,Q \in P({\mathcal{D}})$ and let s(P, Q) denote the number of edges between P and Q in the underlying graph of s. Define the polynomial

\[G_s = \prod_{\{P,Q\}}\lambda_{\{P, Q\}}^{s(P,Q)}.\]

Let $2^{P({\mathcal{D}})}/\sim$ denote the set of unordered bipartitions of the poles of ${\mathcal{D}}$ . (The equivalence relation $\sim$ identifies each subset of $P({\mathcal{D}})$ with its complement.) Each loop constituent of s separates the sphere into two regions and determines an element of $2^{P({\mathcal{D}})}/\sim$ . For each $U \in 2^{P({\mathcal{D}})}/\sim$ , let $n_U(s)$ denote the number of loop constituents of s corresponding to U. Let $n_0(s) \;:\!=\; n_{[\emptyset]}(s)$ denote the number of loops that create a region with no poles. We call such loops nullhomotopic. We define the polynomial $L_s$ in variables $\{x_U \mid U \in 2^{P({\mathcal{D}})}/\sim, U \neq [\emptyset]\}$ by

\[L_s = \prod_{\substack{U \in 2^{P({\mathcal{D}})}/\sim\\U \neq [\emptyset]}}x_U^{n_U(s)}.\]

Fix a set of shortcuts $\{\alpha_{P,Q} \mid P, Q \in P({\mathcal{D}})\}$ between every pair of poles of ${\mathcal{D}}$ , where $\alpha_{P,Q}$ is oriented from P to Q. Suppose e, f are segment constituents, each belonging to either ${\mathcal{D}}$ or one of its states. Given orientations on e and f, define

\[\mu_{e,f}(P,Q) = e \cdot \alpha_{P,Q} - f \cdot \alpha_{P,Q}.\]

We say e and f are aligned and write $e \parallel f$ if their endpoint poles coincide and they are identical (as unoriented constituents) in a neighbourhood of each of the endpoint poles.

Suppose $e \in E({\mathcal{D}})$ is oriented from pole $e_0$ to pole $e_1$ . For a given state s, there is at most one $e_s \in E(s)$ aligned with e. If such an $e_s$ exists, we give it the orientation that agrees with e near $e_0$ and $e_1$ . Define the polynomial $E_s$ in variables $\{y_{e, P} \mid e \in E({\mathcal{D}}), P \in P({\mathcal{D}})\}$ by

\[E_s = \prod_{\substack{e \parallel e_s\\p \in P({\mathcal{D}})}}y_{e, P}^{\mu_{e_s, e}(e_0, P)},\]

where the product is taken over all $e \in E({\mathcal{D}}), P \in P({\mathcal{D}})$ such that s has a segment constituent $e_s \in E(s)$ with $e \parallel e_s$ .

Finally, define the generalised bracket polynomial $\ll {\mathcal{D}} \gg$ as the Laurent polynomial in variables A, $\{\lambda_{\{P,Q\}}\}$ , $\{x_U\}$ , and $\{y_{e, P}\}$ given by

(5·1·1) \begin{equation} \ll {\mathcal{D}} \gg = \sum_{s} A^{\sigma(s)}(\!-\!A^2-A^{-2})^{n_0(s)} \cdot G_s \cdot L_s \cdot E_s.\end{equation}

We show in Lemma 5·1 that $E_s$ is well-defined, independent of the choice of shortcuts $\{\alpha_{P,Q}\}$ . The lemma should be thought of as an analog of Corollary 4·2 for closed (but not generic) curves. We argue similarly to the proof given for lemma 8·1 in [ Reference Turaev31 ]. (See also the shortcut moves of [ Reference Linov27 ].)

Observe that (i) and (ii) each preserve $e \cdot \alpha_{P,Q}$ and $f \cdot \alpha_{P,Q}$ , while (iii) and (iv) each preserve $e \cdot \alpha_{P,Q} - f \cdot \alpha_{P,Q}$ by the alignment condition.

The normalised generalised bracket polynomial $\ll {\mathcal{D}} \gg_*$ is defined by

\[\ll {\mathcal{D}} \gg_* = (\!-\!A^3)^{-\mathrm{wr}({\mathcal{D}})} \ll {\mathcal{D}} \gg.\]

Here, the writhe $\mathrm{wr}({\mathcal{D}})$ of the oriented generalised knotoid diagram ${\mathcal{D}}$ is defined by the sum of $\mathrm{sgn}(c)$ over all crossings c in ${\mathcal{D}}$ .

Proof. Observe that the generalised bracket polynomial satisfies the usual disjoint union and skein relations

\[\ll {\mathcal{D}} \cup \bigcirc \gg = (\!-\!A^2-A^{-2})\ll {\mathcal{D}} \gg \quad \text{and} \quad \ll {\mathcal{D}} \gg = A\ll {\mathcal{D}}_+ \gg + A^{-1}\ll {\mathcal{D}}_- \gg,\]

where ${\mathcal{D}}_+$ and ${\mathcal{D}}_-$ denote the diagrams obtained by performing an A-smoothing and a B-smoothing, respectively, at a particular crossing of ${\mathcal{D}}$ . As with the traditional bracket polynomial for knots, this implies that the normalised polynomial $\ll {\mathcal{D}} \gg_*$ is preserved by Reidemeister moves.

In analogy to Proposition 4·11 for the base-pointed index polynomial, Proposition 5·5 gives lower bounds on heights between poles in terms of the generalised bracket polynomial. Before stating the bounds, we record a lemma.

Lemma 5·4. Let e,f be as in Lemma 5·1 and let $P,Q,R \in P({\mathcal{D}})$ . Then

\[\mu_{e,f}(P,Q) + \mu_{e,f}(Q,R) = \mu_{e,f}(P,R).\]

Proof. If $P = Q$ or $Q = R$ , the result follows from Corollary 5·2, so assume this is not the case. Without loss of generality, we choose $\alpha_{P,Q}$ and $\alpha_{Q,R}$ with no intersections other than poles; by Lemma 5·1, this does not change any of the values in question. Then we obtain a shortcut $\tilde \alpha_{P,R}$ by concatenating $\alpha_{P,Q}$ with $\alpha_{Q,R}$ and perturbing the resulting curve in a neighbourhood of Q so that, locally, it neither passes through Q nor intersects e or f. (This is possible even if e and f have an endpoint at Q since $e \parallel f$ .) Observe that $e \cdot \alpha_{P,Q} + e \cdot \alpha_{Q, R} = e \cdot \tilde \alpha_{P,R}$ and $f \cdot \alpha_{P,Q} + f \cdot \alpha_{Q, R} = f \cdot \tilde \alpha_{P,R}$ , whence

\begin{align*} \mu_{e,f}(P,Q) + \mu_{e,f}(Q,R) &= (e \cdot \alpha_{P,Q} - f \cdot \alpha_{P,Q}) + (e \cdot \alpha_{Q,R} - f \cdot \alpha_{Q,R})\\ &= e \cdot \tilde \alpha_{P,Q} - f \cdot \tilde \alpha_{P,Q}\\ &= \mu_{e,f}(P,R), \end{align*}

where the last equality follows from Lemma 5·1.

Following the notation of [ Reference Turaev31 ], we define the $x_i$ -span, denoted $\mathrm{spn}_{x_i}(F)$ , of a Laurent polynomial $F \in \mathbb Z[x_1^{\pm 1}, \dots, x_n^{\pm 1}]$ as the non-negative difference between the largest and smallest exponents of $x_i$ among monomials with nonzero coefficients in F (by convention, $\mathrm{spn}_{x_i}(0) = -\infty$ ). Given a generalised knotoid $\kappa$ and poles $P, Q \in P(\kappa)$ , we let $\{P \mid Q\}$ denote the set of variables $x_U$ with $U \in 2^{P(\kappa)}/\sim$ such that P and Q are separated by U.

Proof. Let ${\mathcal{D}}$ be any diagram of $\kappa$ and let $\gamma$ be any shortcut oriented from P to Q. Let $d_a$ denote the right side of the inequality in (a). Then there is a state s of ${\mathcal{D}}$ with $d_a = \sum_{U \in \{P\mid Q\}} n_U(s)$ . Since $\gamma$ has at least one intersection with each loop constituent of s that separates P and Q, part (a) follows.

We now prove (b) and (c). First, suppose e, f are oriented segment constituents, each belonging to either ${\mathcal{D}}$ or a state of ${\mathcal{D}}$ . Each point of intersection of $\gamma$ with $e \cup f$ away from poles is counted by $e \cdot \gamma - f \cdot \gamma$ with multiplicity in $\{-2, -1, 0, 1, 2\}$ . It follows that the number of such points, denoted $\#(\gamma \cap (e \cup f))$ , satisfies

(5·1·2) \begin{equation} \#(\gamma \cap (e \cup f)) \ge \frac 12 |e \cdot \gamma - f \cdot \gamma|. \end{equation}

Let $d_b$ denote the right-hand side of the inequality in (b). Suppose $d_b > 0$ . Then there is a segment constituent $e \in E({\mathcal{D}})$ , a state s, and a segment constituent $e_s \in E(s)$ such that $e \parallel e_s$ and

\begin{align*} d_b &= \frac 12 |\mu_{e_s, e}(e_0, P) - \mu_{e_s, e}(e_0, Q)|\\ &= \frac 12 |\mu_{e_s, e}(P, Q)|\\ &= \frac 12 |e_s \cdot \gamma - e \cdot \gamma|, \end{align*}

where the second equality uses Lemma 5·4 and the third uses Lemma 5·1. It follows from Inequality (5·1·2) that $\#(\gamma \cap (e_s \cup e)) \ge d_b$ , and part (b) follows.

Finally, let $d_c$ denote the right-hand side of the inequality in (c). Assume $d_c > d_b$ . Then there is a segment constituent $e \in E({\mathcal{D}})$ , two states $s_1, s_2$ , and two segment constituents $f_1 \in E(s_1)$ , $f_2 \in E(s_2)$ such that $e \parallel f_1 \parallel f_2$ and

\begin{align*} d_c &= \frac 12 |(\mu_{f_1, e}(e_0, P) - \mu_{f_1, e}(e_0, Q)) - (\mu_{f_2, e}(e_0, P) - \mu_{f_2, e}(e_0, Q))|\\ &= \frac 12 |-\mu_{f_1, e}(P, Q) + \mu_{f_2, e}(P,Q)|\\ &= \frac 12 |\mu_{f_2, f_1}(P,Q)|\\ &= \frac 12 |f_2 \cdot \gamma - f_1 \cdot \gamma|, \end{align*}

where the second equality uses Lemma 5·4 and the fourth uses Lemma 5·1. It follows from Inequality (5·1·2) that $\#(\gamma \cap (f_1 \cup f_2)) \ge d_c$ , and part (c) follows.

Example 5·6. Let $\kappa$ denote the generalised knotoid represented by the diagram in Figure 15. Computations yield

\begin{align*} \ll \kappa \gg_* = &-\lambda_{QR}\lambda_{PS}(1 + A^{-2} + A^{-2}x_{[\{T\}]} + A^{-4}) - \lambda_{PR}\lambda_{QS}A^{-4}\\ &-\lambda_{PQ}\lambda_{RS}y_{e,Q}y_{e,R}y_{e,S}y_{f,P}^{-1}(A^{-2}y_{f,T} + A^{-4}x_{[\{T\}]} + A^{-6}y_{e,T}^{-1}). \end{align*}

For brevity, we have omitted the set notation in the subscripts of the $\lambda$ variables. Proposition 5·5(a) yields the height bounds $h_{\kappa}(T, P)$ , $h_{\kappa}(T, Q)$ , $h_{\kappa}(T, R)$ , $h_{\kappa}(T, S) \ge 1$ . Meanwhile, Proposition 5·5(b) and 5·5(c) yield the height bounds $h_\kappa(P, Q)$ , $h_\kappa(P, R)$ , $h_\kappa(P, S) \ge 1/2$ and $h_{\kappa}(T, P)$ , $h_{\kappa}(T, Q)$ , $h_{\kappa}(T, R)$ , $h_{\kappa}(T, S) \ge 1$ . In this case, the bounds give enough information for a complete description of the height spectrum of $\kappa$ .

Fig. 15. An oriented generalised knotoid with five poles P, Q, R, S, T and two constituents e, f.

We remark on the case of generalised knotoids that are not necessarily oriented or fully labelled. Consider an oriented, pole-labelled, constituent-labelled diagram ${\mathcal{D}}$ . As we remove orientation or labelling data from ${\mathcal{D}}$ , we alter the polynomial $\ll {\mathcal{D}} \gg_*$ accordingly, as follows.

1. (i) If constituent labels are removed, we re-index the variables $\{y_{e,P}\}$ by replacing each subscript component $e \in E({\mathcal{D}})$ with the ordered pair $(e_0, e_1)$ corresponding to its endpoint poles.

2. (ii) If pole labels are removed, we remove $G_s$ and $E_s$ from the definition given in Equation (5·1·1), and we re-index the variables $\{x_U\}$ as follows. For each subscript U, write $U = [S]$ for a set $S \subset P({\mathcal{D}})$ , then replace U with the integer $\min(|S|, |P({\mathcal{D}})| - |S|)$ .

3. (iii) If orientation is disregarded, we remove $E_s$ from the definition given in Equation (5·1·1) and redefine the normalised bracket polynomial by $\ll {\mathcal{D}} \gg_* = (\!-\!A^3)^{-\sum_{e \in C({\mathcal{D}})}\mathrm{wr}(e)}\ll {\mathcal{D}} \gg$ , where $\mathrm{wr}(e)$ denotes the sum of the signs of the self-crossings of an arbitrarily-oriented constituent e. (In particular, $\mathrm{wr}(e)$ is independent of orientation.)

Given an oriented, unlabelled diagram ${\mathcal{D}}$ (for example), we obtain the bracket polynomial $\ll {\mathcal{D}} \gg_*$ by labelling ${\mathcal{D}}$ arbitrarily and then applying (i) and (ii). (Note that (ii) and (iii) subsume (i).) It is straightforward to verify that, in general, this method yields a bracket polynomial invariant for each class of generalised knotoids with specified labelling and orientation data.

5·2. Recovering existing bracket polynomials

We survey some existing bracket polynomial constructions for classes of objects subsumed by generalised knotoids and show that they are recovered by the generalised bracket polynomial.

In the case of a classical link diagram L, there are no poles, so all loop constituents of any state s are nullhomotopic and $n_0(s)$ is simply the number of constituents of s. Moreover, $G_s$ , $L_s$ , and $E_s$ are constant, so $\ll L \gg_*$ recovers the traditional bracket polynomial $\langle L \rangle$ of the link diagram.

In [ Reference Gabrovšek and Gügümcü12 ], a bracket polynomial for oriented, ordered (i.e. pole-labelled) multi-linkoids L is defined by

\[\langle L \rangle_\bullet(A, \{\lambda_{ij}\}) = (\!-\!A^3)^{-\mathrm{wr}(L)}\sum_{s} A^{\sigma(s)}(\!-\!A^2-A^{-2})^{\|s\|}\prod_{\Lambda} \lambda_{ij}.\]

Here, the poles of the multi-linkoid are labelled $1, 2, \dots, 2n$ , the symbol $\Lambda$ indicates that the product is taken over all $i < j$ such that poles i and j are connected by a segment constituent in s, and $\|s\|$ denotes the number of loop constituents of s. It is clear that $\langle L \rangle_\bullet$ is recovered from $\ll L \gg_*$ by identifying $G_s$ with $\prod_{\Lambda}\lambda_{ij}$ and making the substitutions $x_U = -A^2 - A^{-2}$ for all $U \in (2^{P({\mathcal{D}})}/\sim)\setminus [\emptyset]$ and $y_{e, P} = 1$ for all $e \in E(L), P \in P(L)$ .

In [ Reference Turaev31 ], Turaev defines a two-variable bracket polynomial for oriented spherical knotoids. Let K be a spherical knotoid oriented from L to H, and let $\alpha = \alpha_{L,H}$ be a shortcut oriented from L to H. (We also use K to denote the segment constituent of the knotoid.) Turaev’s polynomial is given by

\[\ll K \gg_\circ(A, u) = (\!-\!A^3)^{-\mathrm{wr}(K)}\sum_{s}A^{\sigma(s)}(\!-\!A^2 - A^{-2})^{\|s\|}u^{k_s \cdot \alpha - K \cdot \alpha},\]

where $k_s$ denotes the unique segment constituent of the state s, oriented from L to H. Note that for any state s, the segment constituent $k_s$ aligns with K, and all loop constituents of s are nullhomotopic. In terms of the generalised bracket polynomial, the polynomials $G_s = \lambda_{LH}$ and $L_s = 1$ are constant, and $n_0(s) = \|s\|$ . It follows that $\ll K \gg_\circ$ is recovered from $\ll K \gg_*$ by the substitutions $\lambda_{LH} = 1$ and $y_{K, H} = u$ . It also follows that the height bound in Proposition 5·5(c) is equivalent to the complexity bound given in [ Reference Turaev31 , equation 8·3·1].

In [ Reference Turaev31 ], Turaev extends $\ll K \gg_\circ$ to a bracket polynomial for oriented planar knotoids given by

\[[K]_\circ(A, B, u) = (\!-\!A^3)^{-\mathrm{wr}(K)} \sum_s A^{\sigma(s)}(\!-\!A^2 - A^{-2})^{p(s)}B^{q(s)}u^{k_s \cdot \alpha - K\cdot \alpha},\]

where p(s) denotes the number of loop constituents of s that do not enclose the segment constituent $k_s$ and q(s) denotes the number of loop constituents of s that enclose $k_s$ . Represent K as a generalised knotoid with poles $L,H, \infty$ and constituent K oriented from L to H, equipped with shortcuts $\{\alpha_{P,Q} \mid P,Q \in \{L, H, \infty\}\}$ , where $\alpha_{L,H} = \alpha$ . Then $p(s) = n_0(s)$ is simply the number of nullhomotopic loop constituents of s, and $q(s) = n_{[\{\infty\}]}(s)$ is the number of loop constituents of S that determine the bipartition $\{\{\infty\}, \{L, H\}\}$ . Thus $[K]_\circ$ is recoverable from $\ll K \gg_*$ by the substitutions $\lambda_{LH} = 1$ , $x_{[\{\infty\}]} = B$ , $y_{K, \infty} = 1$ , and $y_{K, H} = u$ .

In [ Reference Kutluay26 ], Kutluay refines $[K]_\circ$ by replacing the term $u^{k_s \cdot \alpha - K \cdot \alpha}$ with the term $\ell^{w_{\gamma_s}(L) - w_\gamma(L)}h^{w_{\gamma_s}(H) - w_\gamma(H)}$ , obtaining a polynomial $[K]_{\bullet}(A,B,\ell, h)$ . Here, $w_\beta \colon \mathbb R^2 \to \mathbb{Z}/2$ is the winding potential function of an oriented generic closed curve $\beta$ that maps each point $x \in \mathbb R^2$ to the winding number of $\beta$ around x. (For $x \in \beta$ , the winding number is taken to be the average of the winding numbers of the regions adjacent to x.) The closed curve $\gamma$ (resp. $\gamma_s$ ) is defined by $K \cup \alpha^r$ , (resp. $k_s \cup \alpha^r$ ), where $\alpha^r$ is $\alpha$ with reversed orientation. Kutluay observes that

\[k_s \cdot \alpha - K \cdot \alpha = [w_{\gamma_s}(L) - w_\gamma(L)] - [w_{\gamma_s}(H) - w_\gamma(H)],\]

so Turaev’s polynomial $[K]_\circ$ is recoverable from Kutluay’s polynomial $[K]_\bullet$ via the substitutions $\ell = u$ and $h = u^{-1}$ .

It is well known that for a generic closed curve $\beta$ on the plane and a point $x \not \in \beta$ , the winding number $w_\beta(x)$ is given by $\beta \cdot \alpha_{x, \infty}$ , where $\alpha_{x,\infty}$ is any path from x to the unbounded region of the plane that intersects $\beta$ transversely and away from self-intersections of $\beta$ . Using this fact, it is straightforward to verify that

\begin{align*} w_{\gamma_s}(L) - w_\gamma(L) &= \gamma_s \cdot \alpha_{L, \infty} - \gamma \cdot \alpha_{L, \infty}\\ &= \mu_{k_s, K}(L, \infty),\end{align*}

and similarly

\begin{align*} w_{\gamma_s}(H) - w_\gamma(H) &= \mu_{k_s, K}(H, \infty)\\ &= -\mu_{k_s, K}(L, H) + \mu_{k_s, K}(L, \infty).\end{align*}

It follows that $[K]_\bullet$ is recoverable from $\ll K \gg_*$ by the substituions $\lambda_{LH} = 1$ , $x_{[\{\infty\}]} = B$ , $y_{K, \infty} = \ell h$ , and $y_{K, H} = h^{-1}$ .

6·1. Knotoidal graphs

We now define an extension of generalised knotoids. Let $\Sigma$ denote a closed orientable surface, and let G be a finite graph. As with generalised knotoids, we do not require G to be connected or simple, and G may have valency-zero vertices. Let $\tilde{G}$ denote the disjoint union of G with a finite collection of circles. The edges and circles of $\tilde{G}$ are still called constituents.

A knotoidal graph diagram on $\Sigma$ is a pair $(\mathcal{D}, V(\mathcal{D}))$ , where $\mathcal{D}$ is a generic immersion of $\tilde{G}$ in $\Sigma$ whose only singularities are transverse double points, called crossings, with over/undercrossing data, and where $V(\mathcal{D})$ is a chosen subset of the images of the vertices of G, each of which has valency at least one. We call the elements of $V(\mathcal{D})$ the spatial vertices of $\mathcal{D}$ . The images of the remaining vertices of $\tilde{G}$ are called poles of $\mathcal{D}$ and the set of poles is denoted by $P({\mathcal{D}})$ . Figure 16 shows an example.

Fig. 16. A knotoidal graph with three poles and four spatial vertices.

The rest of the terminology for generalised knotoids carries over. we call the graph G (resp. $\tilde{G}$ ) the underlying graph (resp. underlying looped graph) of $\mathcal{D}$ . The valency of a pole or spatial vertex is the valency of the corresponding vertex in the underlying graph G. Let $E({\mathcal{D}})$ denote the set of images of the edges of $\tilde{G}$ , called segment constituents of $\mathcal{D}$ , and let $L(\mathcal{D})$ denote the images of the circles of $\tilde{G}$ , called loop constituents of ${\mathcal{D}}$ . Let $C({\mathcal{D}}) \;:\!=\; E({\mathcal{D}}) \cup L({\mathcal{D}})$ denote the set of constituents of ${\mathcal{D}}$ .

We consider knotoidal graph diagrams in $\Sigma$ up to ambient isotopy and generalised Reidemeister moves: the three standard Reidemeister moves away from poles and spatial vertices, along with the vertex slide move and the vertex twist move near spatial vertices. See Figure 17. The forbidden pole moves of Figure 1 remain in effect. A knotoidal graph KG is an equivalence class of knotoidal graph diagrams. We may speak of knotoidal graphs possibly with labels assigned to poles or spatial vertices, and labels or orientations assigned to constituents.

Fig. 17. The vertex slide move (top) and the vertex twist move (bottom).

With the vertex twist move, the theory of knotoidal graphs is akin to the theory of spatial graphs with pliable vertices (as opposed to rigid vertices, where the twist move is not allowed). One could require knotoidal graphs to have rigid spatial vertices, which would yield a different theory that we do not consider in this paper.

As we remarked for generalised knotoids, an equivalent theory arises by considering knotoidal graphs in compact surfaces with boundary while disallowing valency-zero poles.

6·2. Rail diagrams

A topological perspective on knotoids is introduced in [ Reference Kodokostas and Lambropoulou25 ] by means of rail diagrams. For planar knotoids, consider the thickened disk $D^2 \times I$ together with two distinct lines $\ell_1 = \{x\} \times I, \ell_2 = \{y\} \times I$ , called rails. A planar rail diagram is a proper embedding of an arc into $(D^2 \times I) \setminus \;{} ^{^{^{\circ\!\!\!\!\!\!}}}\mathring{N}\;(\ell_1 \cup \ell_2)$ such that one endpoint is embedded in $\partial N(\ell_1)$ and one is embedded in $\partial N(\ell_2)$ , as shown in Figure 18. We consider planar rail diagrams up to isotopies of the embedded arc in $(D^2 \times I) \setminus\;{} ^{^{^{\circ\!\!\!\!\!\!}}}\mathring{N}\;(\ell_1 \cup \ell_2)$ . Intuitively, the removal of the rails has the effect of topologically enforcing the forbidden move for knotoid diagrams. It is shown in [ Reference Kodokostas and Lambropoulou25 ] that the theory of planar rail diagrams is equivalent to the theory of planar knotoids, and if the disk $D^2$ is replaced by an arbitrary surface $\Sigma$ , the arguments in [ Reference Kodokostas and Lambropoulou25 ] generalise readily to show that the theory of $\Sigma$ -rail diagrams is equivalent to the theory of knotoids on $\Sigma$ .

Fig. 18. A planar rail diagram for a knotoid.

We extend $\Sigma$ -rail diagrams to knotoidal graphs. For a given knotoidal graph KG in $\Sigma$ with diagram $\mathcal{D}$ , take the thickened surface $\Sigma \times I$ and n distinct rails $\ell_i = \{x_i\} \times I$ , $i = 1, \dotsc, n$ , where $n = |P(\mathcal{D})|$ , each rail corresponding to a pole $p_i$ . Let $\tilde{G}'$ be the graph obtained from the underlying looped graph $\tilde{G}$ of $\mathcal{D}$ by removing a disk neighbourhood of each pole. Then for each pole $p_i$ there are $v(p_i)$ endpoints in $\tilde{G}'$ created by removing the disk neighbourhood. Now properly embed $\tilde{G}'$ in $(\Sigma \times I) \setminus \;{} ^{^{^{\circ\!\!\!\!\!\!}}}\mathring{N}\;(\bigcup_{i = 1}^n \ell_i)$ so that the projection of the embedding to $\Sigma \times \{0\}$ recovers $\mathcal{D}$ away from the poles, and so the $v(p_i)$ endpoints of $\tilde{G}'$ are embedded on the boundary $\partial N(\ell_i)$ of the corresponding rail neighbourhood.

Again, we consider these diagrams up to isotopies of the embedded looped graph in $(\Sigma \times I) \setminus \;{} ^{^{^{\circ\!\!\!\!\!\!}}}\mathring{N}\;(\bigcup_{i = 1}^n \ell_i)$ , with one major difference. We may identify the cylindrical boundary portion of each $\partial N(\ell_i)$ with $S^1 \times I$ . If an endpoint v of $\tilde{G}'$ is embedded as the point (x, t) in $\partial N(\ell_i)$ , then we require the isotopies to keep v in the vertical line segment $\{x\} \times I$ . That is, we only allow endpoints of the graph $\tilde{G}'$ embedded in rail neighbourhoods to slide vertically up and down. This prevents arbitrary twisting near poles, which we disallow in the diagrammatic theory of knotoidal graphs.

The theory of $\Sigma$ -rail diagrams of knotoidal graphs is equivalent to the theory of knotoidal graphs in $\Sigma$ . Consequently, we may derive invariants of knotoidal graphs using topological methods, as we do in Section 7.

Although we do not address it here, one can also allow virtual crossings in a spherical or planar knotoidal graph. This would generate classical knotoidal graphs in surfaces as we have described, however, in this theory, different projections can generate surfaces of different genus and equivalence between diagrams must then allow adding and removing handles for the corresponding surfaces. See [ Reference Gügümcü, Kauffman and Pongtanapaisan19 ] for this theory as applied to graphoids.