[Paper Review] Unknown values in the table of knots

Jae Choon, Charles Livingston|arXiv (Cornell University)|Mar 7, 2005

Geometric and Algebraic Topology16 references3 citations

TL;DR

This paper maintains a dynamic, regularly updated list of prime knots with few crossings for which fundamental invariants—such as unknotting number, smooth 4-genus, and A-polynomial—remain unknown. It leverages collaboration with the KnotInfo database to compile and verify open problems in low-dimensional topology, providing a critical reference for researchers in knot theory.

ABSTRACT

This paper, to be regularly updated, lists those prime knots with the fewest possible number of crossings for which values of basic knot invariants, such as the unknotting number or the smooth 4-genus, are unknown. This list is being developed in conjunction with "KnotInfo" (www.indiana.edu/~knotinfo), a web-based table of knot invariants.

Motivation & Objective

• To identify and compile all prime knots with few crossings for which key topological invariants remain unknown.
• To maintain a living, updatable record of open problems in knot theory, particularly for invariants with no known efficient computation algorithms.
• To support and integrate findings from the KnotInfo database to ensure accuracy and traceability of unresolved values.
• To encourage community contributions by requiring verifiable sources for new entries, ensuring academic rigor.
• To highlight persistent gaps in knowledge for invariants like the smooth 4-genus, unknotting number, and A-polynomial, especially for 12-crossing knots.

Proposed method

• The paper compiles data from the KnotInfo database (www.indiana.edu/~knotinfo), which systematically tabulates known values of knot invariants.
• It uses algorithmic results and computational advances—such as those from Ng, Litherland, and Dynnikov—on arc index, Thurston-Bennequin number, and Turaev genus to update known values.
• For invariants like the A-polynomial, it incorporates results from hyperbolic geometry and gluing equations of ideal tetrahedra in knot complements.
• The authors apply topological invariants such as Heegaard-Floer homology and Donaldson theory to rule out unknotting number 1 for specific knots.
• They maintain a dynamic table of unknown values, using notations like '12?' to indicate that 12-crossing knots are pending computation.
• Contributions are only included when referenced by peer-reviewed papers or verified websites, ensuring reliability.

Experimental results

Research questions

• RQ1 Which prime knots with 10–12 crossings have unknown values for the smooth 4-genus?
• RQ2 For which 12-crossing knots is the unknotting number still undetermined despite recent advances?
• RQ3 What is the status of the A-polynomial computation for non-2-bridge knots beyond 9 crossings?
• RQ4 Which knots have unknown values for the Turaev genus, and can they be resolved using recent cobordism bounds?
• RQ5 What is the current status of the Thurston-Bennequin number for 12-crossing non-alternating knots?

Key findings

• The unknotting number remains unknown for 10_11, 10_47, 10_51, 10_54, 10_61, 10_76, 10_77, 10_79, and 10_100.
• The smooth 4-genus is unknown for 11n_80 and 21 additional 12-crossing prime knots.
• The topological 4-genus is unknown for 12a_244, 12a_810, 12a_905, 12a_1142, 12n_549, 12n_555, and 12n_642.
• The arc index remains unknown for 12n_41, 12n_119, 12n_120, 12n_121, 12n_145, 12n_153, 12n_199, 12n_200, 12n_243, 12n_260, 12n_282, 12n_310, 12n_322, 12n_351, 12n_362, 12n_368, 12n_377, 12n_403, 12n_414, 12n_425, 12n_475, 12n_523, and 12n_549.
• The A-polynomial is unknown for 9_30, 9_32, 9_33, 9_34, 9_39, and 9_40, and remains uncomputed for many 12-crossing knots.
• Lisa Piccirillo’s 2018 result proved that the Conway knot (11n_34) is not smoothly slice, implying its four-genus is 1 and its concordance order is at least 2.