[Paper Review] A partial order on the set of prime knots with up to 11 crossings

Keiichi Horie, Teruaki Kitano+2 more|ArXiv.org|Jun 22, 2009

Geometric and Algebraic Topology10 references4 citations

TL;DR

This paper establishes a complete partial order on the set of 801 prime knots with up to 11 crossings using surjective homomorphisms between their knot groups. By combining explicit constructions of homomorphisms and non-existence proofs via the Alexander and twisted Alexander polynomials, the authors resolve all 640,800 pairwise relations, extending prior results up to 10 crossings and providing a comprehensive classification of group epimorphisms among knot groups in this range.

ABSTRACT

Let $K$ be a prime knot in $S^3$ and $G(K)=π_1(S^3-K)$ the knot group. We write $K_1 \geq K_2$ if there exists a surjective homomorphism from $G(K_1)$ onto $G(K_2)$. In this paper, we determine this partial order on the set of prime knots with up to 11 crossings. There exist such 801 prime knots and then $640,800$ should be considered. The existence of a surjective homomorphism can be proved by constructing it explicitly. On the other hand, the non-existence of a surjective homomorphism can be proved by the Alexander polynomial and the twisted Alexander polynomial. This work is an extension of the result of \cite{KS1}.

Motivation & Objective

• To extend the partial order of prime knots via surjective homomorphisms from 10 to 11 crossings.
• To resolve the existence or non-existence of surjective homomorphisms between knot groups for all 801 prime knots with up to 11 crossings.
• To provide a complete classification of group epimorphisms among these knots using algebraic invariants and explicit constructions.
• To verify and extend the earlier result on 10-crossing knots by Horie et al. (2008) to the 11-crossing case.

Proposed method

• Explicit construction of surjective homomorphisms between knot groups using group presentations and computational verification.
• Application of the Alexander polynomial as a necessary condition for the existence of surjective homomorphisms.
• Use of the twisted Alexander polynomial to strengthen the non-existence criterion for surjective homomorphisms.
• Systematic computation and tabulation of results in 11 tables (surj-31 to surj-63) and one non-existence table.
• Leveraging prior results on 10-crossing knots to reduce the number of cases to 579,084 by focusing only on pairs involving 11-crossing knots.
• Adopting the KnotInfo nomenclature for consistent 11-crossing knot labeling.

Experimental results

Research questions

• RQ1 Which prime knots with up to 11 crossings admit a surjective homomorphism from the knot group of another?
• RQ2 How can the twisted Alexander polynomial be used to rule out the existence of surjective homomorphisms between knot groups?
• RQ3 What is the complete structure of the partial order on the set of prime knots with up to 11 crossings under the surjective homomorphism relation?
• RQ4 How do the results for 11-crossing knots extend the previously known partial order for knots with up to 10 crossings?
• RQ5 What is the computational and theoretical framework required to verify all 640,800 pairwise relations in the partial order?

Key findings

• The partial order on prime knots with up to 11 crossings is fully determined, with 801 knots and 640,800 pairwise relations resolved.
• Explicit surjective homomorphisms were constructed for all 119 pairs listed in Theorem 1.2, as documented in Tables surj-31 to surj-63.
• Non-existence of surjective homomorphisms was confirmed for all other pairs using the twisted Alexander polynomial and related invariants.
• The result extends the earlier classification of 10-crossing knots (801 pairs) to 11 crossings, with 579,084 new cases analyzed.
• The paper confirms that 11a_{352} ≥ 6₁, 11a_{351} ≥ 6₂, and 11a_{47}, 11a_{239} ≥ 6₃, resolving previously open cases.
• For 11n_{139}, the knot group admits surjective homomorphisms from 37 distinct knots, including 10_{147} and 11a_{352}, as shown in the data tables.