Knot Invariants and Strategic Values in Topological Games: A General Framework and New Theorems

1. Introduction

Knot theory studies embeddings of circles and, more generally, finite unions of circles in three-dimensional space, up to ambient isotopy. Diagrams encode these objects by planar projections together with crossing information. The Reidemeister theorem states that two diagrams represent equivalent knots precisely when they are connected by a finite sequence of the three Reidemeister moves [1,2].

Knot invariants, including the Alexander and Jones polynomials, are tools for distinguishing some inequivalent knots [3,4].

There is already a substantial literature on games played on knot and link diagrams. The knotting–unknotting game asks two players to resolve unresolved crossings, with the final winner determined by whether the resulting knot is trivial [5,6]. Adams, Henrich, and Stoll introduced the KnotLink game and proved winning-strategy results for families of rational knots and links [7]. Subsequent work studied Whitehead and pretzel link families [8].

These examples establish that the interaction between knot theory and combinatorial game theory is not new in itself.

The purpose here is narrower. We ask:

Under what conditions can knot or link invariants determine, bound, or distinguish the strategic values of a game played on knot diagrams?

The answer depends on the rules. An invariant of terminal knots may determine terminal payoffs exactly, yet provide too little information to determine which player can force which terminal outcome.

We make this distinction precise using finite game graphs, reachable invariant sets, and quotient constructions.

The main results are general statements about finite games equipped with invariant-valued labels. Their application to a particular knot game requires checking the stated hypotheses against that game’s legal moves and payoff rule.

2. Topological and Game-Theoretic Preliminaries

2.1. Knots, Links, Diagrams, and Invariants

A knot is an embedding $S^1\hookrightarrow S^3$, considered up to ambient isotopy. A link is an embedding of a finite disjoint union of circles into $S^3$, again considered up to ambient isotopy.

A diagram is a generic planar projection with over/under information at each crossing. The Reidemeister theorem provides the diagrammatic equivalence criterion [1,2].

Definition 1.  

Let $\mathcal{T}$ be a collection of knot or link types and let A be a set. A map

$$I:\mathcal{T}⟶A$$

is aninvariantif equivalent diagrams representing the same topological type have the same I-value.

It is acomplete invarianton $\mathcal{T}$ if $I\left(K\right)=I\left(K^{\prime }\right)$ implies $K=K^{\prime }$ as topological types for all $K,K^{\prime }\in \mathcal{T}$.

The Alexander polynomial [3] and Jones polynomial [4] are standard examples of knot invariants, but neither should be treated as a complete invariant for all knots.

For links, the linking number of an oriented two-component link is another classical invariant [9].

Remark 1.  

An invariant is not required to be numerical. Its target may be a set of polynomials, equivalence classes, finite signatures, or tuples of invariants. If an invariant is used only to distinguish the unknot from a nontrivial knot, it may be enough to use a coarse Boolean-valued invariant, provided its values genuinely certify the required distinction.

2.2. Finite Perfect-Information Games

Definition 2.  

A finite two-player deterministic perfect-information game is a tuple

$$G=(P,P_A,P_B,P_T,E,\tau ,u),$$

where P is a finite set of positions, $P_A,P_B$ are the positions at which players $A,B$, respectively, move, $P_T$ is the set of terminal positions, $E\subseteq P\times P$ is the legal-move relation, τ assigns each nonterminal position to its player to move, and $u:P_T\to \mathbb{R}$ is the terminal payoff to player A.

We assume that every play terminates. Player A maximizes payoff and player B minimizes it.

For a finite acyclic game graph, the strategic value is defined recursively by

$$\mathsf{V}\left(p\right)=\left\{\begin{array}{cc}u\left(p\right),\hfill & p\in P_T,\hfill \\ {\displaystyle \underset{q\in E\left(p\right)}{\max }\mathsf{V}\left(q\right),}\hfill & p\in P_A,\hfill \\ {\displaystyle \underset{q\in E\left(p\right)}{\min }\mathsf{V}\left(q\right),}\hfill & p\in P_B,\hfill \end{array}\right.$$

where $E\left(p\right)=\{q:(p,q)\in E\}$.

This is the standard backward-induction or minimax recursion for finite perfect-information games [10,11,12].

In a win–loss game one may take $u\in \{0,1\}$.

Definition 3.  

Aknot-diagram gameis a finite game in which positions encode knot or link diagrams, partial diagrams, or other finite states derived from them, and terminal payoffs depend on the topological type of the terminal diagram.

The crossing-resolution games in [5,7] are motivating examples. A game may also permit smoothing moves, diagram simplifications, or moves between knot and link states, as in the KnotLink game [7].

3. Invariant-Labelled Game Graphs

Let G be a finite knot-diagram game. Let Q be a set of labels and let

$$\lambda :P⟶Q$$

assign a label to every position.

At a terminal position, the label may be the value of a knot invariant of the represented knot or link. At a nonterminal position, $\lambda$ may record an invariant of every possible completion, a set of possible invariant values, or a more detailed state descriptor.

Definition 4.  

For $p\in P$, define the set of terminal descendants

$$T\left(p\right)=\{t\in P_T:\mathrm{there}\mathrm{is}\mathrm{a}\mathrm{directed}\mathrm{path}\mathrm{from}p\mathrm{to}t\}.$$

For an invariant I defined on terminal topological types, let

$${\mathcal{R}}_I\left(p\right)=\{I\left(K_t\right):t\in T\left(p\right)\},$$

where $K_t$ is the knot or link type represented at terminal position t.

We call ${\mathcal{R}}_I\left(p\right)$ thereachable invariant setat p.

This set records what invariant values are possible, but not necessarily which player can force them. It is therefore a useful coarse descriptor, not automatically a strategic value.

Proposition 1  

(Terminal payoff factorization). Suppose the terminal payoff depends only on the invariant I: there is a function $g:A\to \mathbb{R}$ such that

$$u\left(t\right)=g\left(I\left(K_t\right)\right)(t\in P_T).$$

Then terminal payoffs are determined by I. In particular, if $I\left(K_t\right)=I\left(K_{t^{\prime }}\right)$, then $u\left(t\right)=u\left(t^{\prime }\right)$.

Proof. 

The assumed factorization gives

$$u\left(t\right)=g\left(I\left(K_t\right)\right)=g\left(I\left(K_{t^{\prime }}\right)\right)=u\left(t^{\prime }\right).$$

□

Remark 2.  

Proposition 1 concerns terminal payoffs only. It does not imply that equal invariant values at nonterminal positions yield equal minimax values.

4. When an Invariant Determines Strategic Value

4.1. A Sufficient Quotient Condition

A label is useful for strategic analysis when it preserves not just terminal payoffs but also the strategic structure of moves.

Definition 5.  

An equivalence relation ∼ on positions of a finite game is agame congruenceif:

(i) 

if $p\sim q$, then either both are terminal or both are nonterminal;

(ii) 

if $p,q$ are terminal and $p\sim q$, then $u\left(p\right)=u\left(q\right)$;

(iii) 

if $p\sim q$ are nonterminal, then they have the same player to move;

(iv) 

for every move $p\to p^{\prime }$, there is a move $q\to q^{\prime }$ with $p^{\prime }\sim q^{\prime }$, and conversely.

The move-matching condition is understood with respect to the induced quotient game, so that the quotient remains well-founded.

Theorem 1  

(Strategic value descends to a game quotient). Let G be a finite acyclic two-player zero-sum game and let ∼ be a game congruence. Then

$$p\sim q⟹\mathsf{V}\left(p\right)=\mathsf{V}\left(q\right).$$

Consequently, if an invariant label I induces a game congruence by $p{\sim }_{I}q$ if and only if $I\left(p\right)=I\left(q\right)$, then the strategic value is a function of $I\left(p\right)$.

Proof. 

Proceed by induction on the maximum remaining length of a play from a position. If $p,q$ are terminal, the conclusion follows from condition (ii).

Suppose they are nonterminal and equivalent. By condition (iii), the same player moves at both positions. The move-matching condition says that the sets of successor equivalence classes agree. By the induction hypothesis, equivalent successors have equal values.

If player A moves, the maxima over corresponding successor values agree; if player B moves, the minima agree. Thus $\mathsf{V}\left(p\right)=\mathsf{V}\left(q\right)$. □

Remark 3.  

Theorem 1 is a quotient principle for finite game graphs, stated here to clarify what a strategic invariant must preserve. It is not a claim that a familiar knot polynomial automatically satisfies the game-congruence hypotheses.

In most knot-diagram games, a terminal knot invariant alone will not encode the legal moves available from a partial diagram.

4.2. A Counterexample to Terminal-Invariant Sufficiency

Proposition 2.  

There are finite games with identical terminal payoff labels and identical sets of reachable terminal invariant values at two positions, but with different strategic values.

Proof. 

Let the terminal labels be 0 and 1, with terminal payoffs equal to their labels.

At position p, player A moves and has two successors: a terminal position of payoff 0, and a position b at which player B moves and can choose either terminal payoff 0 or terminal payoff 1. Then

$$\mathsf{V}\left(b\right)=\min \{0,1\}=0,\mathsf{V}\left(p\right)=\max \{0,0\}=0.$$

Now let q be a position at which player A moves and has a successor terminal of payoff 1, and a successor position c at which player B moves and has only a terminal successor of payoff 0. Then

$$\mathsf{V}\left(c\right)=0,\mathsf{V}\left(q\right)=\max \{1,0\}=1.$$

Both p and q can reach terminal labels 0 and 1, so their reachable invariant sets are identical, but $\mathsf{V}\left(p\right)\ne \mathsf{V}\left(q\right)$. □

Remark 4.  

This is a game-theoretic counterexample, not a claim about a particular knot family. It proves that a set of possible invariant values does not record the control each player has over those outcomes. A knot application must use additional structural information, such as legal crossing resolutions, move order, and how invariant values evolve.

5. Invariant-Based Bounds on Strategic Values

Suppose the terminal payoff is $g\left(I\right(K\left)\right)$, but I is not a game congruence. The reachable invariant set still yields bounds.

Theorem 2  

(Reachable-set bounds). Let G be a finite terminating two-player zero-sum game. Assume that $u\left(t\right)=g\left(I\left(K_t\right)\right)$ for every terminal position t. For every position p,

$$\underset{a\in {\mathcal{R}}_I\left(p\right)}{\min }g\left(a\right)\le \mathsf{V}\left(p\right)\le \underset{a\in {\mathcal{R}}_I\left(p\right)}{\max }g\left(a\right).$$

Proof. 

Every play from p ends at a terminal descendant $t\in T\left(p\right)$, whose payoff belongs to the set

$$\{g\left(a\right):a\in {\mathcal{R}}_I\left(p\right)\}.$$

Hence every terminal payoff attainable from p lies between the displayed minimum and maximum. A minimax value is obtained by recursively taking maxima and minima of values of successors; these operations cannot move a value outside the interval containing all terminal payoffs reachable from p. Equivalently, apply backward induction to the game tree rooted at p. □

Corollary 1  

(Invariant certificate of a forced outcome). Suppose a win–loss game has terminal payoff 1 exactly when $I\left(K_t\right)\in W\subseteq A$, and payoff 0 otherwise.

(i) 

If ${\mathcal{R}}_I\left(p\right)\subseteq W$, then $\mathsf{V}\left(p\right)=1$.

(ii) 

If ${\mathcal{R}}_I\left(p\right)\cap W=⌀$, then $\mathsf{V}\left(p\right)=0$.

Proof. 

In (i), every terminal descendant has payoff 1, so every play from p has payoff 1. In (ii), every terminal descendant has payoff 0, so every play has payoff 0. In both cases the minimax value is forced. □

Remark 5.  

The corollary is a sufficient condition, not a necessary one. If ${\mathcal{R}}_I\left(p\right)$ contains both winning and losing labels, the game may still have a forced outcome because one player controls the relevant branches of the game tree.

6. Distinguishing Strategic Positions

An invariant can distinguish positions strategically if it separates their values, but equality of invariant values only guarantees equal strategic values under stronger hypotheses.

Theorem 3  

(Value separation). Let $I:P\to A$ be a position label and suppose there is a function $F:A\to \mathbb{R}$ such that

$$\mathsf{V}\left(p\right)=F\left(I\right(p\left)\right)\mathrm{for}\mathrm{every}p\in P.$$

If $F\left(I\right(p\left)\right)\ne F\left(I\right(q\left)\right)$, then p and q have different strategic values. If $I\left(p\right)=I\left(q\right)$, then their strategic values are equal.

Proof. 

Both conclusions follow immediately by substituting the labels into the assumed factorization of $\mathsf{V}$. □

Remark 6.  

The substantive task in applications is to prove the factorization $\mathsf{V}=F\circ I$. Theorem 1 gives one sufficient structural route. In general, the existence of a knot invariant by itself does not establish this factorization.

A practical alternative is to use a vector of features,

$$\mathbf{I}\left(p\right)=\left(I_1\left(p\right),\dots ,I_m\left(p\right),s\left(p\right)\right),$$

where $I_j$ are knot-theoretic labels and $s\left(p\right)$ records game-state information, such as whose turn it is, the number of unresolved crossings, or special moves remaining.

Adding state data can improve prediction, but it proves a theorem only if the resulting label satisfies the quotient conditions or a separate value-factorization argument.

7. Applications to Knot and Link Games

7.1. Knotting–Unknotting Games

In the knotting–unknotting game, players alternately resolve precrossings in a knot shadow, and the terminal objective is to obtain the unknot or a nontrivial knot. Johnson studied this game and analyzed classes of rational shadows and sums of rational shadows [5,6].

These papers show that combinatorial structure and the arrangement of crossings can be decisive.

For this game, let I be a terminal invariant and let W be a set of invariant values certifying nontriviality. If the invariant is sound for the chosen certificate, Corollary 1 can prove a forced result whenever all terminal completions of a position have labels in W, or none do.

A nontriviality certificate may rule out the unknot for a specified terminal diagram. However, failure of a chosen invariant to distinguish a terminal knot from the unknot is inconclusive: it does not prove that the knot is trivial.

7.2. The KnotLink Game

Adams, Henrich, and Stoll introduced the KnotLink game, in which players can resolve crossings and use permitted smoothing moves, and established winning-strategy results for families of rational knots and links [7]. Later work studied families of Whitehead and pretzel links [8].

These results illustrate that useful invariant information often works in combination with local move analysis.

For link positions, linking number can be an effective terminal certificate when its hypotheses apply: for an oriented two-component link, a nonzero linking number implies that the link is not the unlink [9]. This fact can certify a terminal outcome, but it does not by itself prove that a player can force that outcome. The strategic step requires an argument that covers every possible countermove.

8. Conclusions

We have given a general language for studying how knot and link invariants interact with strategic values in finite topological games. The main conclusions are precise but conditional.

A terminal invariant determines terminal payoffs when the payoff factors through it; the set of reachable invariant values yields bounds; an invariant that induces a suitable game congruence determines exact minimax values; and reachable invariant sets alone need not determine strategic value.

The distinction between terminal topology and strategic control is central. A knot polynomial or linking number may certify that a completed diagram has a particular property, while the winner of a game depends on the legal moves and the order in which players control them.

Therefore, the most promising applications combine knot-theoretic certificates with rigorous game-tree arguments. The general propositions here provide a framework for such work, but concrete new classifications and stronger invariant-based theorems remain to be proved for particular game families.