Computations of the slice genus and the unknotting number of links via machine learning

Yutong Dai; Oliver Hayman; András Juhász; Ludovico Morellato

math.GTOct 7, 2026

Computations of the slice genus and the unknotting number of links via machine learning

Authors: Yutong Dai, Oliver Hayman, András Juhász, Ludovico Morellato

Abstract

Links are disjoint unions of circles smoothly embedded in $S^3$ . We use reinforcement learning and Bayesian optimisation to obtain new upper bounds on several link invariants that are not known to be algorithmically computable: the slice genus and the unknotting number for links, and the strong slice genus for algebraically split links. We also compute lower bounds using known invariants. Combining the upper and lower bounds, we obtain new exact values in many cases. Our unknotting agents can reproduce the non-additivity of the unknotting number for several counterexamples due to Brittenham and Hermiller, in some cases finding new unknotting trajectories.

Figures & tables

Figure 1 . A PD code of the trefoil is

[(6,3,1,4),(4,1,5,2),(2,5,3,6)]

, where the crossings are labelled as

Ci

for

i\in\{1,2,3\}

Panel of Figure 2 — Figure 2 . The trefoil has unknotting number at most 1: movie with a band.

Panel of Figure 3 — Figure 3 . Band move. The top strand (in black) is the target edge and the bottom one (in green) is the final part of the band before the band move takes place. Orientation is crucial. On the left, no twist is needed. On the right, we have two possible cases, which we can think of as performing a final Reidemeister I move before the band move takes place. This Reidemeister I move is right-handed in the top picture and left-handed in the bottom picture.

Panel of Figure 4 — Figure 4 . Building a slice surface for the trefoil: first part.

Panel of Figure 5 — Figure 5 . Building a slice surface for the trefoil: second part.

Figure 6 . Schematic of the RL loop.

Tensor Name	Tensor Shape	Tensor Content
PD_code	$c_{\max}\times 4$	PD code of the current diagram
crossing_signs	$c_{\max}$	sign of each crossing
writhe	$1$	writhe of the diagram
gauss_code	$2c_{\max}+\ell_{\max}$	Gauss code
num_link_components	$1$	number of link components that have crossings
unlinked_unknot_components	$1$	number of crossingless unknot components

Table 1. The elements of the base observation common to all frameworks (they describe the diagram and the band being built).

Tensor Name	Tensor Shape	Tensor Content
crossing_changes	$1$	number of crossing changes performed so far (unknotting only)
component_ccl	$\ell_{\max}$	CCL of each link component (genus frameworks only)
genera_info	$\ell_{\max}\times 5$	one row per CCL, recording the CCL, the number of bands attached to that surface component, the number of initial link components (including unknots introduced by birth) and the number of current link components belonging to it, and the number of original link components it contains (genus frameworks only)
current_genus	$1$	genus of the surface built so far (genus frameworks only)

Table 2. The elements of the base observation specific to each framework: crossing_changes is used in the unknotting framework and the last three in the genus frameworks.

Event	$r(a,s^{\prime})$
the diagram becomes terminal (episode won)	$+R_{\text{win}}$
an error state is entered (episode lost)	$-R_{\text{win}}$
a non-terminal end action (a band move or crossing change)	$-5$
an unknot_birth	$-1$
any other action, with pre-action band length $L\geq 9$	$-\min(L-7,\,5)$
otherwise	$-1$

Table 3. The action rewards.

Framework	Training	Validation	Development	Test	Total
unknotting	6310	838	838	841	8827
ribbon , slice	5701	800	801	807	8109
strong ribbon , strong slice	5153	731	868	690	7442

Table 4 . Dataset sizes. The coarse validation set consists of

500

of the validation links.

Framework	Selected feature	Accuracy	Baseline
unknotting	Baseline	$66.8\%$	$66.8\%$
ribbon	DetSig (determinant and signature)	$63.3\%$	$56.4\%$
slice	Seif (Seifert genus upper bound)	$60.2\%$	$57.9\%$
strong ribbon	MT (Murasugi–Tristram bound)	$23.3\%$	$7.7\%$
strong slice	CompM (components matrix)	$14.1\%$	$6.2\%$

Table 5. Selected input feature per framework, with its accuracy on the development set at

200

units per link and, for comparison, the accuracy of the bare state observation. Since each selected accuracy is the best of ten, it is biased upwards. The unbiased figures are those measured on the test set in Section 4.4 .

Framework	p_twist	p_end	$P(\mathrm{end})$	$P(\mathrm{twist})$	max_	max_	p_birth	Accuracy
Framework	p_twist	p_end	$P(\mathrm{end})$	$P(\mathrm{twist})$	twists	actions	p_birth	Accuracy
unknotting	$0.061$	$90.44$	$97.8\%$	$0.07\%$	$1$	$10$	$-$	$58.2\pm 0.3\%$
ribbon	$0.351$	$98.31$	$97.7\%$	$0.35\%$	$1$	$20$	$-$	$56.9\pm 0.6\%$
slice	$0.012$	$86.63$	$97.7\%$	$0.01\%$	$1$	$21$	$0.005$	$56.4\pm 0.2\%$
strong ribbon	$0.023$	$90.02$	$97.8\%$	$0.03\%$	$1$	$17$	$-$	$10.6\pm 0.2\%$
strong slice	$0.022$	$89.80$	$97.8\%$	$0.02\%$	$4$	$17$	$0.001$	$10.7\pm 0.3\%$

Table 6. Tuned random walk parameters per framework, with the probabilities of ending a band and of inserting a twist that they induce. The last column is the validation accuracy, as mean and standard error over five evaluations at

800

units per link.

Framework	Noise floor	Accuracy range of
Framework	Noise floor	p_rw_range	p_twist	p_end	min_p_rw	p_birth
unknotting	$1.10$	$18.3$	$0.32$	$0.69$	$0.61$	$-$
ribbon	$1.10$	$9.4$	$0.91$	$0.80$	$0.89$	$-$
slice	$1.00$	$10.4$	$0.92$	$0.80$	$0.56$	$2.01$
strong ribbon	$1.20$	$12.6$	$0.70$	$0.30$	$0.83$	$-$
strong slice	$0.80$	$6.7$	$0.34$	$0.20$	$0.39$	$0.37$

Table 7 . Accuracy ranges associated with each parameter, computed on the evaluations of the Bayesian optimisation, compared with the noise floor (coarse validation set,

200

units per link). All values are expressed in percentage points.

Framework	p_rw_range	step_decay	min_p_rw	p_birth	Accuracy
unknotting	$(0.70,\,1.00)$	$0.15$	$0.022$	$-$	$86.8\pm 0.3\%$
ribbon	$(0.90,\,1.00)$	$0.15$	$0.006$	$-$	$81.9\pm 0.3\%$
slice	$(0.80,\,0.90)$	$0.30$	$0.0003$	$0.050$	$81.0\pm 0.4\%$
strong ribbon	$(0.75,\,1.00)$	$0.30$	$0.0025$	$-$	$49.0\pm 0.6\%$
strong slice	$(0.70,\,1.00)$	$0.30$	$0.023$	$0.007$	$34.0\pm 0.7\%$

Table 8 . Tuned mixed agent parameters per framework. The accuracy is measured on the validation set at

800

units per link, as mean and standard error over five evaluations with fresh random seeds.

Framework	Most common first band		Distinct first bands per episode
Framework	rl	rl_rw	rl	rl_rw
unknotting	$43\%$	$9\%$	$0.11$	$0.35$
ribbon	$63\%$	$6\%$	$0.12$	$0.64$
slice	$62\%$	$9\%$	$0.13$	$0.68$
strong ribbon	$79\%$	$9\%$	$0.15$	$0.88$
strong slice	$73\%$	$9\%$	$0.18$	$0.89$

Table 9 . Diversity of the attempts of the pure RL policy ( rl ) and of the mixed agent ( rl_rw ) on the development set, at

800

units per link. Most common first band: share of the episodes of a link that open with its most common first band, median over the links. Distinct first bands per episode: number of distinct first bands of a link divided by its number of episodes, averaged over the links.

Framework	Accuracy (%) of				Best agent
Framework	naive_rw	bo_rw	rl	rl_rw	Best agent
unknotting	$8.4\pm 0.8$	$58.6\pm 0.9$	$82.4\pm 0.8$	$\mathbf{87.4\pm 0.7}$	rl_rw
ribbon	$10.2\pm 0.9$	$54.6\pm 1.1$	$73.3\pm 0.4$	$\mathbf{82.1\pm 0.5}$	rl_rw
slice	$9.8\pm 0.5$	$55.2\pm 2.2$	$68.9\pm 0.5$	$\mathbf{80.6\pm 1.1}$	rl_rw
strong ribbon	$1.2\pm 0.3$	$12.7\pm 1.2$	$37.9\pm 1.7$	$\mathbf{51.9\pm 0.8}$	rl_rw
strong slice	$1.4\pm 0.5$	$12.1\pm 0.7$	$27.6\pm 1.3$	$\mathbf{36.7\pm 1.2}$	rl_rw

Table 10 . Accuracy of the four agents on the test set of each framework, expressed as percentages, with mean and standard deviation over five runs at

800

units per link. The highest mean accuracy is in bold.

Framework	Tuning the walker	RW $\rightarrow$ RL	RL $\rightarrow$ RL+RW
Framework	( $\texttt{bo\_rw}-\texttt{naive\_rw}$ )	( $\texttt{rl}-\texttt{bo\_rw}$ )	( $\texttt{rl\_rw}-\texttt{rl}$ )
unknotting	$+50.2$	$+23.8$	$+5.1$
ribbon	$+44.4$	$+18.7$	$+8.8$
slice	$+45.4$	$+13.8$	$+11.7$
strong ribbon	$+11.6$	$+25.2$	$+14.0$
strong slice	$+10.6$	$+15.5$	$+9.2$

Table 11 . Accuracy differences between the various models, in percentage points. Recall that the margin of error on these differences lies between

0.8

and

4.3

points.

Framework	bo_rw steps	Accuracy (%) of			$\texttt{bo\_rw}-\texttt{rl}$	$\texttt{bo\_rw}-\texttt{rl\_rw}$
Framework	per link	bo_rw	rl	rl_rw	$\texttt{bo\_rw}-\texttt{rl}$	$\texttt{bo\_rw}-\texttt{rl\_rw}$
unknotting	$3452$	$81.6\pm 1.0$	$82.3\pm 0.8$	$\mathbf{87.4\pm 0.7}$	$-0.7\pm 1.6$	$-5.9\pm 1.4$
ribbon	$5196$	$\mathbf{84.3\pm 0.9}$	$73.3\pm 0.4$	$82.1\pm 0.5$	$+11.0\pm 2.1$	$+2.2\pm 1.8$
slice	$5725$	$\mathbf{84.4\pm 0.5}$	$68.9\pm 0.5$	$80.6\pm 1.1$	$+15.5\pm 1.9$	$+3.8\pm 1.5$
strong ribbon	$5088$	$29.4\pm 1.0$	$37.9\pm 1.7$	$\mathbf{51.9\pm 0.8}$	$-8.5\pm 4.8$	$-22.5\pm 4.7$
strong slice	$3999$	$25.7\pm 1.5$	$27.6\pm 1.3$	$\mathbf{36.7\pm 1.2}$	$-1.9\pm 4.6$	$-11.1\pm 4.1$

Table 12. The agents on the test set when the tuned random walker is given the computing time of the RL policy. Accuracies are mean and standard deviation over five runs. Differences are in percentage points, with the standard error of the paired comparison link-by-link, allowing for links that share a prime.

	Start			Theoretical contribution		After theory
Invariant	exact	bounded	unknown	made exact	better bounds	exact	bounded
unknotting ${u}(#1)$	$1{,}264$	$0$	$2{,}924$	$1{,}132$	$1{,}792$	$2{,}396$	$1{,}792$
slice genus ${g}_{4}(#1)$	$3{,}781$	$407$	$0$	$19$	$0$	$3{,}800$	$388$
strong slice genus ${g}_{4}^{\ast}(#1)$	$0$	$0$	$4{,}188$	$3{,}456$	$732$	$3{,}456$	$732$

Table 13. At least two-component oriented prime links with up to

11

crossings: contribution of the theoretical bounds.

	After theory		Agent contribution		Final
Invariant	exact	bounded	made exact	better bounds	exact	bounded
unknotting ${u}(#1)$	$2{,}396$	$1{,}792$	$846$	$934$	$3{,}242$	$946$
slice genus ${g}_{4}(#1)$	$3{,}800$	$388$	$125$	$0$	$3{,}925$	$263$
strong slice genus ${g}_{4}^{\ast}(#1)$	$3{,}456$	$732$	$676$	$56$	$4{,}132$	$56$

Table 14. At least two-component oriented prime links with up to

11

crossings: contribution of the agent, on top of Table 13 .

${g}_{4}^{\ast}(#1){}$	$0$	$1$	$2$	$3$	$4$
# links	$4$	$388$	$238$	$44$	$2$

Table 15. The

676

links of finite strong slice genus whose value was determined exactly.

	Start			Theoretical contribution		After theory
Invariant	exact	bounded	unknown	made exact	better bounds	exact	bounded
slice genus ${g}_{4}(#1)$	$1{,}194$	$408$	$45{,}370$	$1{,}671$	$44{,}107$	$2{,}865$	$44{,}107$
unknotting ${u}(#1)$	$19{,}781$	$27{,}191$	$0$	$1{,}097$	$191$	$20{,}878$	$26{,}094$

Table 16. Prime knots with

14

crossings: contribution from the theoretical bounds.

	After theory		Agent contribution		Final
Invariant	exact	bounded	made exact	better bounds	exact	bounded
slice genus ${g}_{4}(#1)$	$2{,}865$	$44{,}107$	$43{,}167$	$914$	$46{,}032$	$940$
unknotting ${u}(#1)$	$20{,}878$	$26{,}094$	—	—	$20{,}878$	$26{,}094$

Table 17. Prime knots with

14

crossings: what the agent contributes on top of Table 16 .

New lower bound source	# knots
$\tau$ and $\nu_{s}$	$957$
positive and $g_{4}=$ upper bound	$327$
Murasugi–Tristram	$4$
total	$1{,}288$

Table 18. New lower bounds on the unknotting number for

14

-crossing prime knots, with source.

${g}_{4}(#1){}$	$0$	$1$	$2$	$3$	$4$	$5$	$6$
# knots	$1{,}194$	$29{,}531$	$10{,}178$	$3{,}822$	$1{,}088$	$202$	$17$

Table 19. The

46{,}032

knots with

14

crossings whose slice genus was determined exactly.

interval	$[0,1]$	$[1,2]$	$[2,3]$
# knots	$10$	$919$	$11$

Table 20. The

940

knots with

14

crossings whose slice genus was narrowed to a short interval, but not determined exactly.

Split	Knot
training	$\mathtt{8_{4}}\sharp\operatorname{m}(#1){\mathtt{9_{25}}}$
	$\mathtt{9_{8}}\sharp\mathtt{9_{38}}$
	$\mathtt{10_{115}}\sharp\mathtt{10_{124}}$
validation	$\mathtt{10_{41}}$
	$\mathtt{10_{41}}\sharp\mathtt{10_{74}}$
coarse validation	—

Table 21 . Rows of the unknotting splits containing

\mathtt{8_{4}}

\mathtt{9_{8}}

\mathtt{9_{10}}

\mathtt{9_{17}}

\mathtt{9_{40}}

\mathtt{10_{41}}

\mathtt{10_{89}}

\mathtt{10_{115}}

\mathtt{10_{121}}

\mathtt{10_{140}}

, or

\mathtt{10_{144}}

(either chirality) as the knot itself or as a connected summand.

Agent	Unknotting sequence
RW	$\mathtt{8_{4}}\sharp\mathtt{9_{10}}$	$\to$	$\mathtt{4_{1}}\sharp\mathtt{9_{10}}$	$\to$	$\operatorname{m}(#1){\mathtt{K15n4866}}$	$\to$	$\mathtt{8_{14}}$	$\to$	unknot
	$[1,4]$ $(2+3)$		$[2,3]$ $(1+3)$		$2$		$1$		$0$
RW	$\mathtt{9_{8}}\sharp\mathtt{9_{10}}$	$\to$	$\mathtt{4_{1}}\sharp\mathtt{9_{10}}$	$\to$	$\mathtt{K15n4005}$	$\to$	$\mathtt{8_{14}}$	$\to$	unknot
	$[3,4]$ $(2+3)$		$[2,3]$ $(1+3)$		$2$		$1$		$0$
RW	$\mathtt{9_{17}}\sharp\mathtt{9_{10}}$	$\to$	$\mathtt{4_{1}}\sharp\mathtt{9_{10}}$	$\to$	$\operatorname{m}(#1){\mathtt{K15n4866}}$	$\to$	$\mathtt{8_{14}}$	$\to$	unknot
	$[3,4]$ $(2+3)$		$[2,3]$ $(1+3)$		$2$		$1$		$0$

Table 22 . The unknotting sequences found by the tuned random walker (RW) and by the mixed agent (RL

+

RW); each arrow is one crossing change. The knot

\mathtt{Kn}

is not in SnapPy’s census up to 15 crossings, found as a diagram with

n

crossings. Knots with the same label

\mathtt{Kn}

in different rows are different as they are hyperbolic and their volumes differ. The knots

\mathtt{K15n4005}

and

\mathtt{K15n4866}

have the chirality of the diagrams SnapPy stores under these names. Below each knot is its unknotting number, or the interval known for it, whose the upper end is the number of crossing changes in the sequence, and whose lower end is the best lower bound of Appendix B . Next to a connected sum, in grey, are the unknotting numbers of its two summands, whose sum is the value that additivity would give.

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