[Paper Review] TQFT computations and experiments

Norbert A’Campo|ArXiv.org|Nov 30, 2003

Geometric and Algebraic Topology6 references3 citations

TL;DR

This paper presents a computational framework, TQFT, for explicitly calculating the projective representation of the mapping class group on Verlinde modules in 3D TQFT. Using Pari-gp and fusion rules from Roberts, Masbaum, and Vogel, the program computes Dehn twist actions via admissible colorings of trivalent graphs, demonstrating that TQFT invariants can distinguish mutant knots with identical classical invariants by detecting non-conjugate monodromy actions via trace invariants in cyclotomic fields.

ABSTRACT

With the help of a new program, we do computations concerning the Witten-Reshetikhin-Turaev representations of mapping class groups. In particular we distinguish some mutant fibered knots. The program can be downloaded from http://www.geometrie.ch/TQFT

Motivation & Objective

• To develop a computational tool for explicit TQFT representations of the mapping class group on Verlinde modules.
• To implement the fusion rules of Roberts, Masbaum, and Vogel for efficient computation in quantum invariants.
• To test whether TQFT invariants can distinguish mutant knots that are indistinguishable by classical invariants.
• To explore the non-conjugacy of monodromy representations in PGL(n, C_{2k+4}) for different knot types.
• To investigate whether TQFT representations detect non-conjugate elements in SL(2,Z) at higher levels.

Proposed method

• The program TQFT uses Pari-gp to compute matrices over the cyclotomic field C_{2k+4} = Q(A)/(Φ_{2k+4}(A)) for Verlinde modules.
• Admissible (k,i)-colorings of trivalent graphs Γ_g are defined by even edge colors satisfying triangular and sum inequalities, with a fixed boundary color i.
• Mapping class group generators (Dehn twists about A_e and B_r) are implemented as matrices via the action on basis elements of the Verlinde module.
• The program initializes via init_so(k) and init_boom_so([0,1,...,g-1],i) to set the level and compute the basis of admissible colorings.
• Matrix actions are computed using twA(e) and twB(r), with results stored in cyclotomic fields for trace analysis.
• Trace invariants of monodromy matrices (up to third powers) are computed and compared to detect non-conjugacy in PGL(n, C_{2k+4}).

Experimental results

Research questions

• RQ1 Can TQFT invariants distinguish mutant knots that are indistinguishable by Kauffman, HOMFLY, and Khovanov invariants?
• RQ2 Do the TQFT representations of monodromy maps for slalom knots yield non-conjugate matrices in PGL(n, C_{2k+4})?
• RQ3 Is there a level k ≥ 3, odd, such that non-conjugate elements in SL(2,Z) remain non-conjugate under TQFT representations?
• RQ4 Can the trace of powers of monodromy matrices detect differences in TQFT representations when lower traces coincide?
• RQ5 Does the TQFT representation of the mapping class group fail to extend to GL(2,Z) due to non-conjugate images of conjugate elements?

Key findings

• For genus g=1 and k=7, the TQFT representation ρ₇: SL(2,Z) → PGL(3,C₁₈) does not extend to GL(2,Z), as the images of conjugate elements in SL(2,Z) are not conjugate in PGL(3,C₁₈).
• The matrices corresponding to a=[7,3;2,1] and b=[7,1;6,1] in SL(2,Z) are not conjugate in PGL(2,C₁₄) at level k=5, despite being conjugate in SL(2,Q).
• For the mutant pair 15n30444 and 15n30419, the TQFT actions on V⁶_{3,0} have equal first and second power traces but distinct third power traces, proving non-conjugacy in PGL(675,C₁₀).
• The computation for the 15n30444/15n30419 pair took 2 minutes 18.84 seconds and produced 675×675 matrices, with trace differences appearing at the third power.
• For the genus 5 slalom knots with i=2, the TQFT actions on V⁵_{3,2} (size 275) have equal first and second power traces but distinct third power traces, confirming non-conjugacy.
• The program successfully distinguishes knots with identical Kauffman, HOMFLY, and Khovanov invariants by detecting non-conjugate monodromy actions via trace invariants in cyclotomic fields.