[Paper Review] Gordian distance and Vassiliev invariants

Sebastian Baader|ArXiv.org|Mar 27, 2007

Geometric and Algebraic Topology6 references3 citations

TL;DR

This paper proves that between any two knots of Gordian distance two, there exist infinitely many knots with arbitrarily prescribed Vassiliev invariants of any finite order. Using special $C_n$-moves and Habiro's theory of $C_n$-moves, the authors construct such knots by manipulating strand permutations and linking numbers in knot diagrams, generalizing Ohyama's result on unknotting number one knots.

ABSTRACT

The Gordian distance between two knots measures how many crossing changes are needed to transform one knot into the other. It is known that there are always infinitely many non-equivalent knots `between' a pair of knots of Gordian distance two. In this paper we prove an extreme generalisation of this fact: there are knots with arbitrarily prescribed Vassiliev invariants between every pair of knots of Gordian distance two.

Motivation & Objective

• To generalize Ohyama's result that every finite set of Vassiliev invariants can be realized by an unknotting number one knot.
• To show that between any two knots of Gordian distance two, there exist infinitely many intermediate knots with prescribed Vassiliev invariants up to any finite order.
• To establish a constructive method for generating such intermediate knots using $C_n$-moves and strand permutation techniques.
• To demonstrate that Vassiliev invariants of order ≤ m can be preserved while altering higher-order invariants via controlled topological moves.

Proposed method

• Construct a knot $K'$ between two knots $K_0$ and $K_1$ of Gordian distance two by introducing a controlled number of half-twists around a negative crossing $C$, altering the linking number of a two-component link $L$.
• Use the skein relation for the Casson invariant ($a_2$) to show that $a_2(K') = a_2(K_0) - lk(L)$, allowing arbitrary adjustment of the Casson invariant via $lk(L)$.
• For higher-order invariants, apply special $C_{m+1}$-moves to modify the cyclic order of strands at a crossing section, enabling control over Vassiliev invariants of order $m+1$.
• Leverage Habiro's theorem that two knots have identical Vassiliev invariants up to order $n$ if and only if they are related by a finite sequence of $C_{n+1}$-moves.
• Use Ohyama and Tsukamoto's result that the change in a Vassiliev invariant of order $n$ under a $C_n$-move depends only on the permutation of strands and the product of crossing signs.
• Modify the strand order via regular isotopy to realize any desired permutation $\sigma \in S_{m+1}$, thereby controlling the effect of $C_{m+1}$-moves on invariants of order $m+1$.

Experimental results

Research questions

• RQ1 Can Vassiliev invariants of arbitrary finite order be prescribed for knots lying between two knots of Gordian distance two?
• RQ2 Is it possible to construct intermediate knots with identical Vassiliev invariants of order ≤ m while maintaining Gordian distance one to both endpoints?
• RQ3 How do $C_n$-moves affect the values of Vassiliev invariants, and can they be used to systematically control these invariants?
• RQ4 To what extent can strand permutation and linking number manipulation in knot diagrams be used to generate knots with specific invariants?
• RQ5 Does the existence of infinitely many intermediate knots between two knots of Gordian distance two extend to control over all finite-order Vassiliev invariants?

Key findings

• For any knot $K$ and any natural number $m$, there exists a knot $K'$ such that $d_G(K', K_0) = d_G(K', K_1) = 1$ and all Vassiliev invariants of $K'$ of order ≤ $m$ coincide with those of $K$.
• The Casson invariant ($a_2$) of $K'$ can be made arbitrarily close to any integer by adjusting the linking number $lk(L)$ of a two-component link derived from a crossing change at a negative crossing.
• By applying a special $C_{m+1}$-move, the knot $K''$ can be constructed such that its Vassiliev invariants of order up to $m+1$ match those of $K$, while preserving distance one to $K_0$ and $K_1$.
• The effect of a $C_n$-move on a Vassiliev invariant of order $n$ depends only on the permutation of strand order and the product of crossing signs, enabling precise control over invariant values.
• The construction is iterative: starting from a knot with controlled $a_2$, higher-order invariants are controlled step-by-step using $C_n$-moves and strand reordering.
• The result generalizes Ohyama’s theorem by showing that not only can finite sets of invariants be realized by unknotting number one knots, but they can also be realized in a controlled way between any two knots of Gordian distance two.