Webring Winding Number (WWN)

For a closed curve \(\gamma\) around point \(z_0\) in the complex plane:

\begin{equation} \text{Ind}_\gamma(z_0) = \frac{1}{2\pi i} \oint_\gamma \frac{d\zeta}{\zeta - z_0} \in \mathbb{Z} \end{equation}

This integer-valued result is a *topological invariant*—it doesn't change under continuous deformation of the path. The formula connects complex analysis (Cauchy integral) with algebraic topology.

The set of homotopy classes of loops on the circle forms a group:

\begin{equation} \pi_1(S^1) \cong \mathbb{Z} \end{equation}

Every loop on the circle is characterized by a single integer: its winding number. Two loops are homotopic (continuously deformable into each other) if and only if they have the same winding number. Counterclockwise is positive, clockwise is negative.

From Wikipedia's example:

This curve has total curvature 6π, turning number 3, though it only has winding number 2 about p.

For a WebRing with ord=7 sites, consider a user navigating:

        site₁ ←──── site₇
         ↓            ↑
       site₂        site₆      Ring<ord=7>
         ↓            ↑
       site₃ ────→ site₅      User path: 1→2→3→2→3→4→5→6→7→1
                ↘   ↗
               site₄

Path analysis for 1→2→3→2→3→4→5→6→7→1:

• Winding (w): 1 — completed one full circuit
• Turning (t): 1 — reversed direction once (at site 3)
• Total steps: 9 — but that's not tracked

The distinction:

• Winding: Complete circuits around the ring (crossing the 7→1 boundary)
• Turning: Direction reversals (changing from next to prev or vice versa)

State after this path: \(\mathcal{C} = \langle \text{ord}=7, k=1, w=1, t=1, d=+ \rangle\)

For a WebRing \(\mathcal{R}\) with \(\text{ord}\) sites (order of cyclic group \(\mathbb{Z}_{\text{ord}}\)), define the navigation state:

\begin{equation} \mathcal{C} = \langle k, w, t, d \rangle \quad \text{where} \quad k \in \mathbb{Z}_{\text{ord}}, \; w \in \mathbb{Z}_{16383}, \; t \in \mathbb{N}, \; d \in \{+, -\} \end{equation}

Transition functions:

• \(\text{next}: \mathcal{C} \to \mathcal{C}'\) where \(k' = (k \mod \text{ord}) + 1\), \(w' = w + \mathbf{1}_{k=\text{ord}}\), \(t' = t + \mathbf{1}_{d=-}\), \(d' = +\)
• \(\text{prev}: \mathcal{C} \to \mathcal{C}'\) where \(k' = ((k-2) \mod \text{ord}) + 1\), \(w' = w - \mathbf{1}_{k=1}\), \(t' = t + \mathbf{1}_{d=+}\), \(d' = -\)

The indicator \(\mathbf{1}_{cond}\) is 1 if condition holds, 0 otherwise. Default state is \(d=+\) (positive/forward).

Consider a webring with ord sites arranged on a circle. The key insight:

   ord = 1:  ●──────────────────●   (self-loop)
                      ↺
   ord = 3:  ●───●───●───●───●  (triangle, but still $S^1$)
                ↺
   ord = ∞:  ●●●●●●●●●●●●●●●●●●  (dense points, still $S^1$)
                      ↺

The number of sites is irrelevant to the topology. All of these are homeomorphic to \(S^1\). The winding number w tracks complete circuits, not individual hops.

When ord = 1 (a ring of one), every navigation is a wraparound:

Action	Position Change	Winding Change
next	k=1 → k=1	w → w+1
prev	k=1 → k=1	w → w-1

You're spinning in place, but the winding number still accumulates. This is mathematically identical to walking in a circle—even a circle of radius zero has well-defined winding.