Walker Anubhav Jain XX XX Month 20XX

# Extracting Structured Seed-Mediated Gold Nanorod Growth Procedures from
Literature with GPT-3

Nicholas Walker  Energy Technologies Area, Lawrence Berkeley National
Laboratory, 1 Cyclotron Road, Berkeley, 94720, CA, United States of America
John Dagdelen  Energy Technologies Area, Lawrence Berkeley National
Laboratory, 1 Cyclotron Road, Berkeley, 94720, CA, United States of America
Department of Materials Science and Engineering, University of California
Berkeley, 210 Hearst Memorial Mining Building, Berkeley, 94720, CA, United
States of America  Kevin Cruse  Materials Sciences Division, Lawrence Berkeley
National Laboratory, 1 Cyclotron Road, Berkeley, 94720, CA, United States of
America  Department of Materials Science and Engineering, University of
California Berkeley, 210 Hearst Memorial Mining Building, Berkeley, 94720, CA,
United States of America  Sanghoon Lee  Energy Technologies Area, Lawrence
Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, 94720, CA, United
States of America  Department of Materials Science and Engineering, University
of California Berkeley, 210 Hearst Memorial Mining Building, Berkeley, 94720,
CA, United States of America  Samuel Gleason  Energy Technologies Area,
Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, 94720, CA,
United States of America  Department of Chemistry, University of California
Berkeley, 419 Latimer Hall, Berkeley, 94720, CA, United States of America
Alexander Dunn  Energy Technologies Area, Lawrence Berkeley National
Laboratory, 1 Cyclotron Road, Berkeley, 94720, CA, United States of America
Department of Materials Science and Engineering, University of California
Berkeley, 210 Hearst Memorial Mining Building, Berkeley, 94720, CA, United
States of America  Gerbrand Ceder  Materials Sciences Division, Lawrence
Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, 94720, CA, United
States of America  Department of Materials Science and Engineering, University
of California Berkeley, 210 Hearst Memorial Mining Building, Berkeley, 94720,
CA, United States of America  A. Paul Alivisatos  Materials Sciences Division,
Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, 94720, CA,
United States of America  Department of Materials Science and Engineering,
University of California Berkeley, 210 Hearst Memorial Mining Building,
Berkeley, 94720, CA, United States of America  Department of Chemistry,
University of California Berkeley, 419 Latimer Hall, Berkeley, 94720, CA,
United States of America  Kavli Energy NanoScience Institute, University of
California Berkeley, 101C Campbell Hall, Berkeley, 94720, CA, United States of
America  Kristin A. Persson  Molecular Foundry, Lawrence Berkeley National
Laboratory, 1 Cyclotron Road, Berkeley, 94720, CA, United States of America
Department of Materials Science and Engineering, University of California
Berkeley, 210 Hearst Memorial Mining Building, Berkeley, 94720, CA, United
States of America  Kavli Energy NanoScience Institute, University of
California Berkeley, 101C Campbell Hall, Berkeley, 94720, CA, United States of
America  Anubhav Jain  Energy Technologies Area, Lawrence Berkeley National
Laboratory, 1 Cyclotron Road, Berkeley, 94720, CA, United States of America
<EMAIL_ADDRESS>
###### Abstract

Although gold nanorods have been the subject of much research, the pathways
for controlling their shape and thereby their optical properties remain
largely heuristically understood. Although it is apparent that the
simultaneous presence of and interaction between various reagents during
synthesis control these properties, computational and experimental approaches
for exploring the synthesis space can be either intractable or too time-
consuming in practice. This motivates an alternative approach leveraging the
wealth of synthesis information already embedded in the body of scientific
literature by developing tools to extract relevant structured data in an
automated, high-throughput manner. To that end, we present an approach using
the powerful GPT-3 language model to extract structured multi-step seed-
mediated growth procedures and outcomes for gold nanorods from unstructured
scientific text. GPT-3 prompt completions are fine-tuned to predict synthesis
templates in the form of JSON documents from unstructured text input with an
overall accuracy of $86\%$. The performance is notable, considering the model
is performing simultaneous entity recognition and relation extraction. We
present a dataset of 11,644 entities extracted from 1,137 papers, resulting in
268 papers with at least one complete seed-mediated gold nanorod growth
procedure and outcome for a total of 332 complete procedures.

###### keywords:

Natural Language Processing, GPT-3, AuNR, Synthesis, Seed-mediated, Named
Entity Recognition, Relation Extraction

##  1 Introduction

Gold nanoparticles have been synthesized for centuries due to their
interesting optical properties, dating back to the Lycurgus Cup from 4th
century Rome,[aunp_hist_0] as well as imperial bowls and decorated dishes from
the Qing dynasty.[aunp_hist_1] However, scientific interest did not develop
until the work of Michael Faraday in the mid-19th century, when he
accidentally synthesized colloidal gold while investigating the interaction
between light and matter.[aunp_hist_2] In the last three decades, chemists
have developed the ability to synthesize anisotropic metal nanoparticles in a
controllable and reproducible fashion.[aunr_hist] Around the turn of the
millennium, multi-step seed-mediated growth methods were developed to prepare
gold nanorods with aspect ratios ranging from $8$ to $20$.[aunr_hist,
aunr_synth, aunr_synth_2] This generated a great deal of interest in
anisotropic gold nanoparticles due to a combination of the convenience of the
wet-chemistry approach, as well as the ability to tune the shape of the
synthesized nanorods. The anisotropic gold nanoparticles, in turn, provide
access to shape-dependent optical phenomena not observed with spherical gold
nanoparticles.[aunr_optical_0, aunr_optical_1, aunr_optical_2, aunr_optical_3]
Their applications are widespread across many domains, including semiconductor
technology,[aunp_semiconductor, aunp_semiconductor_2]
biomedicine,[aunp_biomedical, aunp_biomedical_2] and
cosmetics.[aunp_cosmetics] The suitability of a nanoparticle for a particular
application depends on its morphology and size, which correspond to different
plasmonic properties.[aunp_size_morph, aunp_size, aunp_size_shape]

Despite the popularity of anisotropic gold nanoparticles, systematic
investigation of the control of these properties has only recently been
approached.[aunp_syst] Although some theories and models do exist for
identifying and explaining the mechanisms of synthesis that determine
nanoparticle morphology,[aunr_hist, aunp_form_model, aunp_shape_control_0,
aunp_shape_control_1] most synthesis exploration is still guided by heuristics
based on domain knowledge.

For gold nanorods, it is clear that the simultaneous presence of various
reagents during the synthesis affects the characteristics of the resulting
gold nanoparticles.[aunr_hist] To better understand these effects,
computational simulation and analysis of the formation energetics of the
nanoparticles or the nucleation and growth steps can be used. Density
functional theory (DFT) can be used to investigate the energetic landscape of
potential gold nanoparticle morphologies, including the effects of surface
ligands that are vital for the solution-phase synthesis of noble metal
nanoparticles.[aunp_dft, aunp_ligand_0, aunp_ligand_1] However, this approach
does not account for the nuances of nucleation and growth competition in
solution-based nanoparticle syntheses. These aspects can be addressed by
modeling real-time growth and dispersity dynamics with continuum-level model,
though this sacrifices access to small-scale energetics granted by
DFT.[aunp_continuum] Alternatively, direct experimentation can be used to
explore the synthesis space by varying precursor amounts over many
experiments, though this is impractical due to the both the number of
experiments required to sample the synthesis space and the condition that a
single experiment can take many hours to complete. Automated labs may address
this problem in the future, though most are still in their infancy.

A third approach seeks to leverage the wealth of information contained in
scientific literature. Many seed-mediated gold nanorod recipes have been
published in the materials science and chemistry literature, but parsing them
requires domain experts to manually read these articles to retrieve the
relevant precursors, procedures, laboratory conditions, and target
characterizations. This comes with its own complications, however, as over
time, the body of materials science literature has grown to an unwieldy
extent, preventing researchers from absorbing the full breadth of information
contained in established literature or even reasonably following research
progress as it emerges.[nlp_matsci_mining_1] Thus, it is unreasonable to
expect domain experts in gold nanoparticle synthesis to manually read and
parse the complete existing synthesis literature efficiently, motivating the
development of high-throughput text-mining methods to extract this
information.

The resulting databases built with these methods are the first steps toward
developing data-driven approaches to understanding synthesis, which are being
developed at an accelerating pace as a rapidly emerging third paradigm of
scientific investigation. Generally speaking, these approaches involve the use
of both conventional and machine learning methods to both build large
databases and perform downstream analysis and inference over said databases.
Natural language processing (NLP) has been successfully applied in the
chemical, medical, and materials sciences to produce structured data from
unstructured text using methods and models such as pattern recognition,
recurrent neural networks, and language models.[matsynth_corpus,
ner_chemistry_2, ner_chemistry_3, ner_medical, ner_biomedical, bioelectra,
weston_ner, matsynth_corpus, nlp_matsci_mining_0, nlp_matsci_0, nlp_matsci_1,
nlp_matsci_2, nlp_matsci_3, nlp_matsci_4, nlp_chem_transfer, nlp_chem_ner_0,
nlp_chem_ner_1, nlp_chem_ner_2, nlp_chem_ner_3, nlp_chem_ner_4,
nlp_chem_ner_5, nlp_biomed_ner_0, matbert_ner]

For applications specifically related to materials synthesis, data-driven
approaches have been successful for tasks such as materials discovery,
synthesis protocol querying, and simulation and interpretation of
characterization results.[tm_mat_disc_0, tm_mat_disc_1, tm_protocol,
tm_char_0, tm_char_1] However, these approaches are fundamentally limited by
the quality of the data, such as the completeness and substance of the data
source. To address this, careful data curation is necessary, as seen with the
construction and maintenance of large databases of characteristic features of
nanostructures.[nano_db]

Recently, the wealth of unstructured information about gold nanoparticle
synthesis and characterization in literature has been directly tapped through
the combination of various NLP models and other text-mining techniques to
produce a dataset of over five thousand codified gold nanoparticle synthesis
protocols and outcomes.[aunp_synth_dataset] This general dataset contains a
wealth of information, including detected materials, material quantities,
morphologies, synthesis actions, and synthesis conditions, as well as tags for
seed-mediated synthesis, synthesis paragraph classifications, and
characterization paragraph classifications.

Despite the breadth of accurate information provided, the general dataset
still suffers from a few pitfalls: (i) the inability to distinguish between
seed and growth solution procedures in seed-mediated growth synthesis; (ii)
the inability to detect references to materials that do not contain specific
formulae or chemical names (e.g. “AuNP seed solution”); and (iii) the
inability to detect target morphologies as opposed to incidentally mentioned
morphologies. To address these issues, this work intends to use a large
sequence-to-sequence language model to extract full synthesis procedures and
outcomes in a single-step inference. Generally speaking, a sequence-to-
sequence model in NLP maps an input sequence to an output sequence by learning
to produce the most likely completion of the input by conditioning the output
on the input.[s2s]

In this work, we leverage the capabilities of the latest language model in the
Generative Pre-trained Transformer (GPT) family, GPT-3,[gpt3] to build a
dataset of highly structured synthesis templates for seed-mediated gold
nanorod growth. A similar approach using GPT-3 to build materials science
datasets has been applied to extracting dopant-host material pairs, cataloging
metal-organic frameworks, and extracting general
chemistry/phase/morphology/application information for materials.[gpt3_mastci]
We extracted these templates for seed-mediated gold nanorod growth from 2,969
paragraphs across 1,137 papers, starting with using a question-answering
framework aided by the zero-shot performance of GPT-3 to construct a small
initial dataset. We then fine-tuned GPT-3 to produce complete synthesis
templates for input paragraphs. Fine-tuning GPT-3 consists of using multiple
examples of paragraph and synthesis template pairs to train GPT-3 to perform
this specific task. Each synthesis template in the final dataset contains
information on relevant synthesis precursors, precursor amounts, synthesis
conditions, and characterization results, all structured in a JSON format.
This dataset provides reproducible summaries of procedures and outcomes,
explicitly establishing the relationships between the components of the recipe
(e.g. accurately linking the correct volumes and concentrations with the
correct precursors in the correct solution). However, this specificity comes
at the cost of generality, as the dataset focuses on seed-mediated gold
nanorod growth. The final dataset consists of 11,644 entities extracted from
1,137 papers, 268 of which contain least one complete seed-mediated gold
nanorod growth procedure and outcome for a total of 332 complete procedures.

##  2 Dataset

The relevant data for constructing the training, testing, and prediction data
for this model was collected using the database of gold nanoparticle synthesis
protocols and outcomes developed by Cruse et al.[aunp_synth_dataset] from the
full-text database developed by Kononova et al.[matsynth_corpus] through text-
and data-mining agreements with several major scientific journal publishers.
The original full-text database contains more than 4.9 million materials
science articles, and the pipeline for identifying and extracting gold
nanoparticle synthesis articles consists of progressively finer-meshed
filtering steps using text-mining tools adapted from Kononova et
al.[matsynth_corpus] and Wang et al.[sol_synth_dataset] These steps include
regular expression matching to identify nanomaterial papers, document and
vocabulary vectorization using term frequency-inverse document frequency (TF-
IDF) to reveal papers related more to gold than other noble metals, BERT-based
binary classifiers to identify paragraphs related to gold nanoparticle
synthesis or characterization (particularly morphological information), a
combination of BiLSTM-based named entity recognition (NER) and rules-based
methods to extract synthesis procedure details from synthesis paragraphs, and
MatBERT[matbert_ner] NER to extract morphology and size information from
characterization paragraphs.

Using the extracted information, 5,145 papers were identified to contain gold
nanoparticle synthesis protocols,[aunpdataset] of which 1,137 papers were
found to contain seed-mediated recipes using the "seed_mediated" flag as well
as rod-like morphologies ("rod or "NR" in "morphologies" under
"morphological_information") or aspect ratio measurements ("aspect" or "AR" in
"measurments" under "morphological_information"). This was done to filter the
total papers down to only those likely to contain seed-mediated synthesis
recipes for gold nanorods.

##  3 Methods

At the core of the GPT-1 model was a focus on improving language understanding
by generative pre-training involving the use of a large language model in
conjunction with a very large and diverse pre-training corpus with long
stretches of contiguous text, which greatly facilitated the model’s ability to
learn “world knowledge” alongside its ability to process long-range
dependencies.[gpt1] For a sequence-to-sequence generative model, outputs are
generated by maximizing the log probability of
$p(\textrm{output}|\textrm{input})$.[s2s] To further improve zero-shot
performance for both learning and task transfer, GPT-2 modified the training
objective to include task conditioning,
$p(\textrm{output}|\textrm{input},\textrm{task})$, thus establishing the model
as an unsupervised multitask learner.[gpt2] With GPT-3, more extensions of the
model size and the pre-training corpus have produced a model with considerable
capacity for few-shot learning that is capable of producing text that is
difficult to distinguish from human-written text or performing tasks it was
not explicitly trained on, such as writing code or summing numbers.[gpt3] We
employed the 175 billion parameter variant of GPT-3 (OpenAI Davinci) for this
work.

Of the 1,137 papers identified to contain information about seed-mediated gold
nanorod synthesis, 240 (consisting of 661 relevant paragraphs) were randomly
sampled and fully annotated with JSON-formatted recipes by a single annotator
with machine assistance to serve as a training set. An additional 40 papers
(consisting of 117 paragraphs) were used for prediction and corrected to serve
as a testing set. Each relevant paragraph was separately annotated due to
length constraints imposed by GPT-3, which limits the capability to process an
entire article at once. A limit of 2,048 tokens is shared between the input
prompt and the output completion, corresponding to approximately 1,500
words.[gpt3]

###  3.1 Overall Procedure

Figure 1: A diagram illustrating the overall procedural approach for
extracting synthesis templates from text with GPT-3 is shown. All unstructured
text paragraphs were drawn from the seed-mediated gold nanorod growth dataset
of 1,137 papers (purple). The first stage involves filling initial templates
using a zero-shot question/answer framework with GPT-3, which is then
corrected (orange). The plus sign indicates a combination of the texts and
queries used as input. Template correction is done through manual editing of
the templates. These corrected templates are used to fine-tune an initial
GPT-3 model, which produces complete templates in a single prediction (green).
From there, the process of iteratively predicting more templates with a fine-
tuned model, correcting them, adding them to the training set, and then fine-
tuning the model again is then performed (blue). The plus signs for these
stages indicate that text-template pairs are used as input for fine-tuning.

A diagram outlining the general process for producing the final fine-tuned
model for template-filling is shown in Figure 1. In the initial stage
(orange), a simple question-answering framework is used to individually fill
in templates for an initial set of paragraphs. These results are then
corrected and used as an initial training set for fine-tuning GPT-3 to produce
complete templates in the second stage (green). The final stage is an
iterative training process in which new templates are predicted, corrected,
and added to the training set to update the fine-tuned model, thus improving
its performance with each iteration. All input texts for each stage are drawn
from the original dataset of synthesis text filtered down to paragraphs likely
to describe seed-mediated gold nanorod growth (purple). Default settings
through the OpenAI API are used for all fine-tunes of the GPT-3 Davinci model,
and a temperature of zero is used for all model predictions with a double line
break as the stop sequence.

###  3.2 Template Structure and Annotation Scheme

Figure 2: A diagram representing the structure of the seed-mediated gold
nanorod growth JSON template. From left to right, the structure is divided
into three components, the seed solution, the growth solution, and the
resulting gold nanorods. For the seed and growth solution components, there
are entries for the precursors and their associated quantities, as well as
entries for experimental conditions such as the age and aging temperatures of
the solutions and stir rates when adding the reducing agent (for the seed
solution) or the seed solution (for the growth solution). For the gold nanorod
component, there are entries for the characterization information that may be
present, including the aspect ratio (ar), length (l), and
longitudinal/transverse surface plasmon resonances (l/tspr).

The structure and content of the synthesis templates are shown in Figure 2.
The synthesis templates are stored as JSON documents, which contain three
components: the seed solution, the growth solution, and the resulting
nanorods. For the seed and growth solutions, the precursors and their
associated volumes (vol), concentrations (concn), and/or masses are recorded,
as well as the ages of the respective mixed solutions at the time of use and
the temperatures (temp) at which they are aged. Furthermore, the stirring
rates when adding sodium borohydride (NaBH4) to the seed solution and when
adding the seed solution to the growth solution are recorded. The shape and
size of the gold seeds in the seed solution are also noted. For the gold
nanorods (AuNR), the aspect ratios (ar), lengths (l), widths (w), and
longitudinal/transverse surface plasmon resonances (SPRs) are recorded. The
JSON documents have identical structures and thus contain an entry for every
value that can be requested; any values not present in a given paragraph are
filled with an empty string.

When available, numerical quantities with units are extracted. For precursor
volumes, the units are provided in variations of liters, though the
concentrations may be measured in either molarity, molality, or weight
percentage. In some cases, the total volume of a collection of precursors may
be specified instead of the individual volumes of the precursors. In this
case, the explicit volume is associated with the first precursor and the
volumes for the remaining precursors refer to the name of the first precursor,
implicitly communicating a shared volume. For temperatures, degrees Celsius
are most commonly provided, though more qualitative descriptions such as “room
temperature” will still be recorded if the explicit temperature is not
provided in the text but a qualitative description is. Similarly, for solution
ages, minutes or hours are most common, but sometimes only descriptions like
“overnight” are provided and recorded. For stirring rates, the revolutions per
minute (rpm) is preferred, but many papers will instead provide descriptions
such as “gentle” or “vigorous” that are recorded. For the gold nanorod
properties, aspect ratios are unitless while the other quantities (length,
width, SPRs) are provided in units of length, with the exception of some cases
where the LSPR is only provided as “NIR” (near-infrared). Throughout all
stages of the annotation process, three additional researchers were consulted
to reach a consensus on the appropriate annotations for various edge cases
caused by unclear wording or other ambiguities.

###  3.3 Question Answering Completions

Figure 3: An example of a question answering completion using GPT-3. The input
is bounded by a purple box containing the prompt (orange), paragraph text
(green), and query (blue). The output is bounded by a red box.

Unfortunately, the standard pre-trained GPT-3 Davinci model is not capable of
providing consistent completed templates of high quality in one request.
However, the model is capable of answering simple questions about synthesis
paragraphs without any fine-tuning, which allows for the fields of the
synthesis templates to be individually filled using answers from a simple
question-answering framework using GPT-3. An example is shown in Figure
3.[example_text] This machine-assisted annotation approach avoids the
laborious process of manually filling in each field of the templates by hand,
as an annotator only needs to verify and correct the provided answers as-
needed. However, this approach does not scale well to large numbers of papers,
as each query is a separate model request, meaning that each paragraph in each
paper would require a large number of requests in order to fill a single
template. Therefore, this approach is used to construct an initial dataset
consisting of synthesis templates for paragraphs from a small number of
papers. Due to the small number of papers used, this initial dataset does not
necessarily capture the variety of precursors or manners in which critical
data can be communicated in text. Nevertheless, these initial templates, when
corrected, provide a suitable starting point for fine-tuning GPT-3 to provide
complete synthesis templates in single requests for each paragraph. Through an
iterative process of fine-tuning GPT-3 on the available templates, predicting
new templates, correcting them, and fine-tuning a new model using all of the
corrected templates constructed thus far, a final fine-tuned model can be
obtained.

The initial synthesis template dataset was constructed using the zero-shot
question-answering framework with 40 randomly sampled papers. If a relevant
precursor, condition, or characterization was identified with regular
expression pattern matching in the paragraph, the framework would be to
request the information using GPT-3. For example, if “ascorbic acid”, “AA”,
“vitamin C”, or “C6H8O6” appeared in the paragraph, the framework would
request the volume, concentration, and mass of ascorbic acid. This initial
dataset only requested information about the eight most common precursors,
including “HAuCl4”, “CTAB”, and “NaBH4” for the seed solution, and “HAuCl4”,
“CTAB”, “AgNO3”, “AA”, and “seed solution” for the growth solution. To capture
different ways of expressing each precursor, multiple aliases were checked to
include variations on chemical names as well as the chemical formulas.
Additionally, the framework requested information about the stir rate when
adding NaBH4 to the seed solution, the age of the seed solution, the
temperature of the seed solution during aging, the size and shape of the
seeds, the stir rate when adding the seed solution to the growth solution, the
age of the growth solution, and the temperature of the growth solution during
aging. All request completions for each paragraph were aggregated into a
single JSON entry according to the synthesis template scheme shown in Figure
2.

The approach of using zero-shot GPT-3 question answering requests to fill the
templates tended to produce poor results, but it offered an acceptable
starting point for collecting structured recipes. Most of the templates only
required correcting the incorrect entries, rather than filling them in
manually from scratch, which greatly accelerated the creation of the initial
dataset. However, some entries had to be added from scratch due to recipes
including precursors outside the initial set of eight common precursors. Note
that the static nature of the synthesis templates across all paragraphs means
that when one paragraph requires the addition of a new precursor to the
template, this is applied to all templates for all paragraphs. Additionally,
annotation was done strictly, requiring that the synthesis method must be
seed-mediated growth and the target gold nanoparticle morphology must be
nanorods. This provides an important test for the model, as the difference
between recipes that produce very similar morphologies can sometimes be
subtle.

###  3.4 Fine-tuning Procedure and Dataset Construction

These corrected templates derived from the question answering completions
provided an initial training set for fine-tuning GPT-3 to produce the desired
filled templates. From there, templates for paragraphs from 40 more randomly
sampled papers were iteratively predicted, corrected (adding new precursors as
necessary), and added to the training set until templates for paragraphs from
240 papers had been corrected in total. With each iteration, the correction
process became much easier and faster. Initially, templates for information-
dense paragraphs took approximately 4 minutes to validate and correct,
whereas, by the end of the process, they took around a minute each. This is
because GPT-3 largely predicted filled templates with high accuracy. The
testing dataset was composed of paragraphs from an additional random sampling
of 40 papers. Not all of the papers filtered from the original dataset were
guaranteed to contain information that should be placed into synthesis
templates. For example, seed-mediated growth or nanorod measurements and
morphologies may only be incidentally mentioned in a given paragraph that is
otherwise not relevant to a specific seed-mediated gold nanorod growth
procedure. Of the 240 papers in the training set and the 40 papers in the
testing set, 141 and 23 papers respectively contained at least one paragraph
with information that could be placed into a synthesis template.

##  4 Results

The described training dataset of synthesis templates was used to fine-tune a
GPT-3 model to reproduce said synthesis templates from the unstructured text.
Default parameters for the fine-tuning process were employed, incurring a cost
of 85.30 USD. The predictions over the testing dataset (40 papers composed of
117 paragraphs) took around eighty minutes to complete and incurred a cost of
14.39 USD. The performance of the fine-tuned model was then evaluated using
the corrected testing dataset.

###  4.1 Error Evaluation Examples and Definitions

Figure 4: A model prediction example is shown, with empty entries omitted. The
original unstructured text is shown on the left, and the components of the
predicted synthesis template in JSON form are shown on the right. The
important information from the unstructured text is colored in orange (for
precursors) and green (for quantities), while any errors are highlighted in
red.

An example prediction is depicted in Figure 4.[example] Errors are highlighted
in red. For this example, two errors were made. First, the quantities for
“Borohydride” in the seed solution were instead placed under “NaBH4” in the
seed solution. Arguably, this is not truly an error since sodium borohydride
is often conventionally referred to as “borohydride”, possibly indicating
“world knowledge” exhibited by GPT-3. However, there are technically other
borohydrides, such as potassium borohydride, that can be used as a reducing
agent for seed-mediated gold nanorod growth,[synth_kbh4] so this was still
marked as incorrect due to possible ambiguity. The second error was the
failure to extract the HCl volume. Note the rather complex relationship in the
growth solution precursor volumes, where CTAB, HAuCl4, ascorbic acid, AgNO3,
and HCl all share the same 25 mL volume. To avoid confusion, the volume is
explicitly associated with the first-mentioned precursor in the mixture, and
the following precursors refer back to that first precursor. This ensures that
downstream applications can unambiguously process the data to mean that the
precursors are sharing a single volume. Other than these two errors, the model
performs very well at extracting quantities in this example.

Figure 5: A diagram depicting the different types of prediction errors made by
the model is presented. Generally, two categories of errors exist: placement
errors and transcription errors. Placement errors refer to whether the
prediction has placed any information, correct or incorrect, into the
appropriate fields as determined by the ground truth. These are indicated with
the lines connecting the fields in the ground truth and the prediction
templates. A false positive prediction occurs when the prediction places
information in a field that is empty, while a false negative prediction is the
reverse. A true negative prediction is when a field is empty in both the
ground truth and the prediction, and a true positive prediction is when a
field is non-empty in both the ground truth and the prediction. Since the
placement evaluations do not consider whether the predicted value in a field
is actually correct for true positives, an additional transcription evaluation
is used to measure how well the predicted value explicitly matches the ground
truth value. These are indicated with boxes encapsulating the fields. The
transcription evaluation is only applied to true positive placements.

For the 117 testing paragraphs, two types of errors are tracked: placement
errors and transcription errors. This is done in order to evaluate the model’s
capability for separately identifying which fields of the synthesis templates
should contain information, as well as how accurate the appropriately placed
information is. To evaluate information placement, only the existence of
information in the fields of the prediction and ground truth synthesis
templates are considered. For example, if the same field contains information
(as opposed to being empty) in both templates, that is considered a true
positive prediction regardless of whether the information explicitly matches.
If both fields are empty, then that is a true negative. If the prediction
field contains information while the ground truth field is empty, then that is
a false positive, while the reverse is a false negative. These categories of
placement errors are used to calculate the precision, recall, and F1-score for
information placement. Examples of these evaluations are shown in Figure 5.

For evaluating transcription accuracy, only the agreement between the
prediction and the annotation for true positive placements are considered, as
the other types of errors are accounted for by the evaluations of information
placement. For numerical values with units, the units must be exactly correct
and the quantitative relative error was calculated according to the function
$s(p,q)=2\cdot\min(p,q)/(p+q)$, which is derived from the absolute
proportional difference $r(p,q)=\absolutevalue{p-q}/(p+q)$ and is bounded on
$[0,1]$ for non-negative $p$ and $q$. Some values may have modifiers attached,
such as “$>3$ h”. If the prediction misses this information, e.g., gives “3
h”, the prediction is considered half-correct even if the quantity and unit
are both correct. Some quantities will additionally be expressed as a range or
list of values. In these cases, the range boundaries are split into a list as
necessary, and the transcription accuracies are scored and aggregated across
the values in the list with proper ordering enforced. For non-numerical
predictions such as stir rates described as “vigorous” or gold seed
morphologies, an exact string match is required for the prediction to be
marked as correct. The combined accuracy is presented as the product of the
F1-score for information placement and the transcription accuracy.

###  4.2 Model Performance

Table 1: Model F1-scores and accuracies for recipe entities aggregated by recipe component. The support numbers in parentheses account for only the true positives used for the accuracy calculation. | Placement | Transcription | Combined |
---|---|---|---|---
| Precision | Recall | F1 | Accuracy | Adj. F1 | Support
Seed Solution | 0.97 | 0.92 | 0.94 | 0.95 | 0.90 | 159 (142)
Growth Solution | 0.90 | 0.94 | 0.92 | 0.96 | 0.88 | 244 (206)
AuNR | 0.79 | 0.74 | 0.76 | 0.95 | 0.72 | 96 (59)
Overall | 0.90 | 0.90 | 0.90 | 0.96 | 0.86 | 499 (407)

The total performance of the model aggregated over each recipe component as
well as all entries is shown in Table 1. The model appears to be proficient at
generally identifying which information should be filled in the template based
on the content of the text, with a rather high F1-score of $90\%$ that favors
neither precision nor recall. It additionally performs exceptionally at
accurately transcribing the information to provide an impressive overall score
of $86\%$. This indicates a significant improvement over comparable efforts in
solid-state synthesis text-mining, which report an overall accuracy of $51\%$
for extracting all recipe items (chemistry, operations, and attributes of the
operations).[matsynth_corpus] Direct comparison is, however, rather
challenging, as some aspects of the two-step, seed-mediated growth synthesis
are more complicated, such as the presence of two solutions with distinct
precursor sets and a greater amount of precursor information needed due to the
solution-based format. On the other hand, solid-state synthesis extraction
carries its own challenges, considering the greater variation in procedural
steps and conditions that must be considered.

It is clear that the F1-scores for the recipe entities associated with the
seed and growth solutions are very promising, indicating that the model is
reliable for extracting the necessary information from the text for the
component solutions to the synthesis procedure. However, the performance is
worse overall for the gold nanorod properties, with an adjusted F1-score of
approximately $72\%$. This is still an improvement over established results,
however, as the gold nanoparticle synthesis protocol and outcome database
developed by Cruse et al.[aunp_synth_dataset] extracts morphology
measurements, sizes, and units with F1-scores of $70\%$, $69\%$, and $91\%$
via NER with MatBERT, without linking them together. This would inevitably
introduce additional sources of error and performance would be additionally
constrained by the lowest performing extractions.

Table 2: Model performance for precursor detection in the seed and growth solution information. | Seed Solution | Growth Solution
---|---|---
Precision | Recall | F1 | Support | Precision | Recall | F1 | Support
Precursor | 0.98 | 0.90 | 0.94 | 61 | 0.93 | 0.92 | 0.92 | 118

Table 2 shows the model performance for detecting precursors in the seed and
growth solutions. Precursor detection is calculated implicitly based on which
precursors the extracted volumes, concentrations, and masses are associated
with. This is a clear improvement over the results in the gold nanoparticle
synthesis protocol and outcome database developed by Cruse et
al.,[aunp_synth_dataset]. The prior work detected precursors via a BiLSTM-
based NER model with an F1-score of $90\%$ without distinguishing between seed
or growth solution precursors and additionally without the capability for
detecting precursors that do not contain specific formulae or chemical names,
such as the seed solution that is added to the growth solution. The fine-tuned
GPT-3 model missed cases where cationic surfactant, PP, BH4, and AuCl3 were
used as well as a case where HCl was used in the seed solution. None of these
cases occurred in the training set. Notably, the model correctly normalized
“AsA” to “AA”, despite “AsA” never appearing in the training data.

###  4.3 Full Dataset

Table 3: A table depicting the format of each data record for each article in the dataset is presented (constructed by merging paragraph templates). The “doi” key contains the article DOI and the “text” key contains index keys of the relevent paragraphs within that article which in turn contain the paragraph text. The “seed” and “growth” keys respectively contain the keys for the seed and growth solution information, including the “prec” key for precursors, the “stir” key for stir rates (when adding the reducing agent for the seed solution and when adding the seed solution for the growth solution), the “temp” key for the aging temperature, and the “age” key for the solution aging time. The “seed” key has an additional “seed” key that contains the “size” and “shape” keys for the size and shape of the seeds in the seed solution. The “prec” key for each solution contains multiple keys for each precursor in each solution, anonymized as “<precursor name>” in the table. For each precursor, there are three keys: “vol”, “concn”, and “mass” for the precursor volume, concentration, and mass, respectively. The “AuNR” key contains keys for measurements of gold nanorod dimensions: “ar”, “l”, “w”, “lspr”, and “tspr” for the aspect ratio, length, width, LSPR, and TSPR, respectively. Each extracted value is additionally stored as a key with a corresponding list of the paragraph indices that the value was extracted from in order to preserve information about entity sources. The final column displays the total number of entities extracted for each key (with no subkeys). Root Key | First Subkey | Second Subkey | Third Subkey |  Description | Total
---|---|---|---|---|---
doi |  |  |  |  Article DOI | 1137
text | <integer> |  |  |  Paragraph text for <integer>th paragraph | 2969
seed | prec | <precursor name> | volume |  Seed solution precursor volume | 1347
concentration |  Seed solution precursor concentration | 1385
mass |  Seed solution precursor mass | 6
seed | size |  |  Seed solution seed size | 137
shape |  |  Seed solution seed shape | 24
stir |  |  |  Seed solution reducing agent stir rate | 266
temp |  |  |  Seed solution aging temperature | 284
age |  |  |  Seed solution aging time | 352
growth | prec | <precursor name> | volume |  Growth solution precursor volume | 2664
concentration |  Growth solution precursor concentration | 2178
mass |  Growth solution precursor mass | 65
stir |  |  |  Growth solution reducing agent stir rate | 134
temp |  |  |  Growth solution aging temperature | 322
age |  |  |  Growth solution aging time | 464
AuNR | ar |  |  |  Gold nanorod aspect ratio | 587
l |  |  |  Gold nanorod length | 443
w |  |  |  Gold nanorod width | 452
lspr |  |  |  Gold nanorod LSPR | 357
tspr |  |  |  Gold nanorod TSPR | 177

The fine-tuned GPT-3 model was applied to the full dataset of 1,137 papers
(2,969 paragraphs) at a total cost of 384.31 USD over 33 hours. In total,
11,644 entities were extracted from the paragraphs that contained information
of interest. The dataset is presented as a JSON file containing a list with
each element corresponding to a single article. Table 3 summarizes the
structure of the JSON documents for each paper alongside a breakdown of how
the total extracted entities across the entire dataset are distributed across
the entity types. While the template extractions were performed paragraph-by-
paragraph, the templates have been merged by article for convenience. However,
this does mean that some conflicts and repetitions are present in the dataset.
A conflict arises when a particular entity type in a paper (e.g. the volume of
a particular precursor) is specified with different values across multiple
paragraphs and a repetition arises when it is specified with the same value
across multiple paragraphs. Of the 11,644 extracted entities, 10,098 ($\sim
87\%$) are uniquely identified, meaning there are no conflicts or repetitions
(the associated value is extracted from exactly one paragraph). An additional
353 entries present at least one conflict without any repetitions, 251 with at
least one repetition and no conflicts, and 57 with both conflicts and
repetitions. Repetitions do not need to be manually resolved since this arises
from the specification of identical information across multiple paragraphs
(e.g. mentioning the gold nanorod aspect ratios in paragraphs about both the
synthesis procedure as well as the nanorod characterization), but conflicts
can be challenging to resolve in a consistent manner without manual
inspection. For instance, if two separate volumes for a particular precursor
are provided in two separate paragraphs, it can be ambiguous whether the
volumes are part of the same synthesis procedure or distinct synthesis
procedures in the same paper due to the lack of cross-paragraph context. With
this in mind, of the 11,644 extracted entities, 10,349 ($\sim 89\%$) can be
safely extracted by automatically resolving repetitions and discarding
entities with conflicts. Of the entities with conflicts, 341 have two distinct
values, 47 have three, 12 have five, 9 have four, and 1 has five.

Table 4: A table depicting the format of each extracted value in the post-processed version of the dataset. Key | Structure |  Description
---|---|---
mod | <modifier> |  A string indicating if a value is a range, approximate, bounding, or unprocessed.
val | [<value>, …, <value>] |  A list of the extracted values. Ranges will consist of two values for the range boundaries. Processed values will be numbers while unprocessed values will be strings.
unit | <unit> |  The units for the extracted values, if applicable, as a string.
src | [[<index>, …, <index>], …, […]] |  A list of lists of paragraph indices to indicate the source for the extracted information.
index | [[<index>, …, <index>], …, […]] |  A list of lists of positional indices to retain ordering for values that were split from a list during post-processing.

With post-processing applied (as was done for evaluation of the testing
dataset), splitting lists of extracted values into distinct entities and
resolving repetitions of identical information extracted across different
paragraphs within the same papers results in a total of 11,770 unique
entities. In the post-processed version of the dataset, each property contains
a list of dictionaries with structures indicated in Table 4.

###  4.4 Full Dataset Analysis

####  4.4.1 Procedure Completeness Analysis

Figure 6: A diagram showing the proportional overlaps of papers with complete
synthesis procedure and outcome components. Each vertex of the triangle
corresponds to the labeled recipe component. The areas of the circles are
proportional to the corresponding number of papers inscribed. The circles on
the midpoints of the edges correspond to papers with complete recipe
components corresponding to the bounding vertices. The center circle
corresponds to the papers with complete recipes and complete
characterizations.

An ideal database of gold nanorod growth procedures should contain fully-
specified, reproducible procedures alongside their outcomes. This is desirable
because missing information could inhibit downstream applications that need
complete information about the synthesis procedure. For instance, if a
scientist wants to reproduce an experiment that produces gold nanorods of a
particular aspect ratio, they would at the very least need to know all of the
relevant seed and growth solution precursors with their amounts. Similarly, a
data science project that intends to investigate the relationship between
procedures and outcomes will need complete information for the seed and growth
solutions in addition to the gold nanorod measurements in order to produce
reliable predictions. To evaluate the completeness of the information this
dataset contains, we examined 1,137 papers in the full prediction dataset. Of
these, 701 (62%) contained at least one paragraph with a non-empty synthesis
template. Of these 701 papers, 678 (97%) fully specified at least one
synthesis component: the seed solution, the growth solution, or the gold
nanorod dimensions. This is encouraging since the vast majority of the papers
that contain information at least fully specify one component of the procedure
or the outcome.

In order to evaluate the completeness of the components of the procedure and
the outcome, for seed and growth solutions, only fully specified precursors
were considered necessary for reproducibility. Auxiliary information, such as
stirring rates, aging times, aging temperatures, and seed particle
morphologies and sizes, while useful, was not considered necessary. The
precursor information was considered to be full specified for a given paper if
all of the precursor quantities were fully specified with either volume and
concentration, mass, or a specific concentration within another solution for
each precursor with extracted quantities. Exceptions were made for water and
the seed solution that is added to the growth solution, which both only needed
a reported volume or mass. Additionally, seed solution in the growth solution
precursors was required for the growth solution precursors to be considered
complete. For the gold nanorod dimensions to be considered complete, either
the aspect ratio, length, or LSPR measurement had to be specified, with the
latter two at least providing an avenue for estimation of the aspect ratio if
reported alone.

Figure 6 shows how the papers in the full prediction dataset are distributed
across fully-specified synthesis procedure and outcome components according to
these criteria. The vast majority of the papers reported gold nanorod
dimensions, with 80% of the 678 papers with at least one fully specified
synthesis component containing fully-specified gold nanorod dimensions.
Additionally, the majority of the papers fully-specified the seed and growth
solutions (respectively 61% and 67%). However, they are distributed such that
40% (268) of the papers fully specified all three components. This is a
reasonable result considering that many papers will directly report the
relevant gold nanorod dimensions without specifying a synthesis procedure,
opting instead to reference the established recipe that the researchers used
to produce the gold nanorods. Additionally, some researchers will opt to
purchase gold seed solution instead of producing their own, which accounts for
cases where some papers are missing information about seed solution
preparation. Most of the papers with fully-specified synthesis procedures and
outcomes (162) used the typical 8-precursor synthesis and an additional 49 use
the same synthesis precursors with the addition of HCl in the growth solution.
In the post-processed version of the dataset, it is determined that of the 268
papers that fully specified all three components, 233 contained exactly one
procedure. An additional 16 contained two, 13 contained three, 3 contained
four, 2 contained five, and 1 contained six for a total of 332 complete
procedures. This final dataset should be suitable for downstream analysis and
inference, given the overall model performance for extracting complete
synthesis procedures and outcomes from the literature.

####  4.4.2 Data Consistency Analysis

Figure 7: A diagram showing the relationships between the gold nanorod aspect
ratios and other gold nanorod measurements extracted from the literature
including the (a) ratio between length and width and (b) the LSPR peak. The
inlier datapoints are shown in purple and the outlier datapoints in red. The
linear regressions derived from the text-mined data using all of the available
data and only the inlier data are respectively shown in red and purple on each
sub-diagram. For the comparison to the ratio between length and width (a), the
ideal relation is shown with a dashed black line and for the LSPR comparison
(b), a simulated relationship is shown with a dashed black line.[ar_lspr_0]

Figure 7 shows the relationship between various measurements extracted from
text compared to the aspect ratios extracted from the text. Only co-occurring
measurements explicitly present within the extracted information from a given
paragraph are considered data points for comparison. No derived measurements
were used. As a sanity check, the first diagram (a) shows the relationship
between the ratios of the explicit lengths and widths present in the text
(excluding ranges) and the reported aspect ratios. Ideally, the relationship
should be an identity as shown with the dashed line. However, while the vast
majority of the data approximately complies with this trend, there are several
outliers that produce deviation from the ideal trend in the regression of the
text-mined data. This is primarily caused by two papers with mismatches in
measurements extracted from three-step seed-mediated gold nanorod overgrowth
procedures where the dimensions of the nanorod seeds used for overgrowth into
nanowires are confused with the dimensions of the nanowires themselves. With
all outliers removed via outlier detection using an elliptic envelope followed
by manual verification, the linear regression almost exactly matches the ideal
relationship. The most common errors were caused by nanorod overgrowth
measurements taken from three-step seed mediated growth procedures and cases
in which the ordering of the aspect ratios and the lengths and widths were
mismatched (e.g. the lengths and widths are listed while the aspect ratios are
presented as a range). Only 8 of the 78 data points were identified as
outliers. For the comparison between the LSPR peaks and the aspect ratios (b),
a strong linear trend is similarly present. However, for this relationship,
there is an additional comparison to a relationship derived from simulation
using a set refractive index for gold nanorods shown in blue, which is in
general agreement with the relationship derived from text-mined empirical
data.[ar_lspr_0] The deviations can be explained by multiple factors including
deviations from ideal conditions shifting the LSPR peaks such as deviation
from spherical end-cap geometries, low nanorod yields, or impurities in the
gold nanorod solution or the nanorods themselves that change the refractive
index (including poor cleaning or high concentrations of silver in procedures
using AgNO3).[ar_lspr_0, ar_lspr_1] While there are extraction errors present,
outlier removal using an elliptic envelope followed by manual verification
does not significantly change the linear regression. Outliers were most
commonly caused by extraction errors that swapped the LSPR and TSPR
measurements provided in the text. Only 9 of the 86 data points were
identified as outliers. Deviation from the theory in such a manner is to be
expected when considering empirical data from real-world experiments. Still,
the LSPR for spheres should be around 520 nm while the text-mined trend line
points towards a value closer to $580\sim 590$ nm. However, for larger aspect
ratios, the text-mined trend line is more representative of the text-mined
empirical data than the trend line derived from simulation. The major outlier
present in the text-mined data is once again explained by a mismatch in
measurements from a three-step seed-mediated gold nanorod overgrowth
procedure.

####  4.4.3 Gold Nanorod Aspect Ratio Distribution Analysis

Figure 8: A diagram showing the distributions of gold nanorod aspect ratios
resulting from different precursor sets including the (a) standard procedure,
(b) the addition of HCl in the growth solution, and (c) all complete precursor
sets. Negligible contributions for aspect ratios larger than 20 are not shown
($P(AR>12)<0.02$). In each sub-diagram, the median is shown with a solid black
line and the first and third quartiles are shown with dashed black lines.

Figure 8 shows the distributions of the aspect ratios extracted from fully-
specified experiments using precursor sets found in more than 10 papers in the
full prediction database (Figures 8a and 8b), in addition to the complete set
of papers (Figure 8c). For many of the papers, the aspect ratios were directly
reported. However, there are multiple different ways that they are reported
that must be addressed in order to properly construct the distributions. If
the aspect ratio is provided as a range of values, the distribution across
that range was taken to be a normal distribution with a mean and standard
deviation determined by the midpoint and endpoints of the range, respectively.
For papers that did not report aspect ratios directly, length and width
information was used instead. In cases where the lengths and widths were
presented as ranges, they were similarly cast as normal distributions, and the
distributions of the aspect ratios were calculated as ratio distributions. For
cases where only the LSPR was provided, the text-mined linear relationship
with outliers removed shown in Figure 7 was used to estimate the aspect
ratios. In cases where any quantities were accompanied by an approximation
modifier (e.g. $\sim$), the values were cast as uniform distributions over the
range of $\pm 10\%$ of the value. Any calculated aspect ratios that fell below
1 (e.g. due to overlaps in length and width distributions for gold nanorods
with small aspect ratios) were inverted.

From the distribution of the standard recipe, it is readily apparent that the
median nanorod aspect ratio is $3.3$ with respective first and third quartiles
of $2.75$ and $3.98$. Comparing with experiments reporting that varying the
concentration of AgNO3 in the growth solution varies the resulting nanorod
aspect ratios from $1.83$ to $5.04$,[ar_range] the distribution of gold
nanorod aspect ratios text-mined from the literature is consistent with this
range, though it is narrower. Notably, there is a non-negligible amount of
samples with aspect ratios greater than 5 in the distribution for the standard
procedure. This is not consistent with heuristic knowledge of the limitations
of the standard procedure for producing large aspect ratio gold nanorods,
usually due to shorter growth times compared to procedures that adjust the pH
of the growth solution to retard the nanorod growth.[growth_limits,
limited_growth] This is primarily due to erroneous extractions of nanowire
measurements from overgrowth experiments or missed precursors based on manual
inspection of the data. However, the statistics are still dominated by the
lower aspect ratios. Comparing to the distribution for experiments using HCl
in the growth solution, it is apparent that the addition produces a
distribution shifted towards larger aspect ratios. This is consistent with
experiments that have determined that the use of HCl in the growth solution
grants broader tunability of the gold nanorod aspect ratios, allowing for more
controlled growth of longer nanorods relative to the standard
procedure.[hcl_growth_0, hcl_growth_1] Notably, $\sim 7\%$ of the procedures
using the standard procedure and $\sim 9\%$ of the procedures using HCl in the
growth solution provide nanorods with aspect ratios of 5 or higher. However,
when all recipes are considered, it is clear that even longer nanorods can be
synthesized, though these recipes are not as popular in the literature.

##  5 Discussion

Overall, the model performs well at identifying and extracting relevant
information specific to seed-mediated gold nanorod growth procedures in the
literature. The model achieves an overall adjusted F1-score of $86\%$ on the
testing dataset, indicating that it performs rather well at the task of
simultaneous entity recognition and relation extraction. However, due to the
static nature of the relations provided by the synthesis template and the
single inference step, the entity recognition and relation extraction tasks
are not easily disentangled, which limits direct comparison with conventional
two-step approaches. Instead, the model performance for information retrieval
is evaluated according to its ability to place information into fields of the
template where information should exist and then the accuracy of the
information that is correctly placed. For information placement, the
precision, recall, and F1-score are balanced at $90\%$, indicating notable
performance with no preference for false positives or false negatives. Of the
information that is correctly placed in the templates, the model predicts the
specific values with $96\%$ accuracy. Thus, the primary source of error is the
accurate placement of information into the template rather than the accurate
prediction of correctly placed information. However, the template model
struggles with identifying new precursors that were not present in the
training set.

The dataset produced by the model provides a wealth of information about seed-
mediated gold nanorod growth experiments and, to our knowledge, constitutes
the largest structured database with this level of depth and completeness. The
model’s ability to distinguish between precursors in the seed and growth
solutions provides an example of very useful information. The simultaneous
identification of precursors alongside linking them to the appropriate
solutions in the two-step seed-mediated procedure had proven difficult using
established methods due to the propagation of errors introduced by the
reliance on separate models for entity extraction and relation. However, with
this model, if a researcher wants to quickly find papers that used a
particular precursor in the seed solution for seed-mediated growth of gold
nanorods, this task can be accomplished with high fidelity using the predicted
templates. Access to this information can be expected to greatly improve tools
for scientific literature searches, as conventional simple keyword searches do
not offer this specific relational dependence for complicated multi-step
procedures.

For a more ambitious goal, the full synthesis procedure data can be leveraged
for multiple downstream tasks, which would require the creation of additional
models for inference. One example would be a model that predicts gold nanorod
dimensions conditioned on a specific synthesis procedure:
$p\pqty{\textrm{properties}|\textrm{procedure}}$. Such a model may be
leveraged to predict the outcomes of proposed procedures without the need to
perform them explicitly. Building on this, the inverse problem,
$p\pqty{\textrm{procedure}|\textrm{properties}}$, can also be modeled. This
would be very useful for streamlining synthesis experiments, as the necessary
procedures for synthesizing gold nanorods with the desired properties can be
inferred to provide a starting point that reduces the number of experiments
that must be conducted to synthesize the desired gold nanorods. However, in
the most likely case, any model trained on literature data alone will be
incomplete and require further data generation and fine tuning.

Furthermore, it is worth considering how these templates fit into a larger
project for downstream synthesis outcome predictions and synthesis procedure
recommendations. The data extracted from literature can be used to pre-train
models used for these purposes, while explicit experimental data can be used
to further train the models to produce better predictions. The new templates
provided by the experimental results are expected to be of extremely high
quality, which will mitigate the errors present in the pre-training data from
literature over time as more experimental results are added to the template
database.

While this dataset is restricted to seed-mediated gold nanorod growth, the
flexibility and performance of the templating approach using GPT-3 motivates
application to other tasks for structured information retrieval from
unstructured scientific text as has been shown in recent
literature.[gpt3_mastci] To this end, the dataset can be extended to
accommodate seed-mediated growth of other gold nanoparticle morphologies,
which may even improve overall model performance, as many errors were caused
by the model erroneously extracting information from procedures that mentioned
nanorod morphologies, but synthesized a different morphology. Additionally,
more complex synthesis methods, such as three-step processes in which nanorods
are first synthesized via seed-mediated growth to be used as seeds in a growth
solution for overgrowth into nanowires, as well as other synthesis methods,
such as citrate reduction, may require the creation of new templates and fine-
tuning a separate model for each synthesis method to improve overall
performance. Generally, it can be expected that more complex templates will
require more examples for fine-tuning.

##  6 Conclusions

The presented model for static structured templating of seed-mediated gold
nanorod growth procedures extracted from unstructured text using GPT-3 is
demonstrated to be a promising approach for constructing high-quality
structured databases of information from the scientific literature. This
approach for extracting seed-mediated gold nanorod procedures and outcomes
achieves an impressive adjusted F1-score of $86\%$ for the simultaneous
identification and linking of synthesis procedure components. We present a
final dataset of 11,644 entities extracted from 1,137 papers, resulting in 268
papers with at least one complete seed-mediated gold nanorod growth procedure
and outcome for a total of 332 complete procedures. This method can
potentially be utilized for many downstream applications including procedure
searches oriented around specific features, statistical analysis of synthesis
outcomes, synthesis outcome predictions conditioned on procedures, and
synthesis procedure recommendations conditioned on outcomes among others given
the wealth of structured information present. Overall, we present this
approach as a flexible candidate for general-purpose structured data
extraction from unstructured scientific text and contribute a dataset that may
serve as a useful tool for investigating synthesis pathways beyond heuristics.

## Data and Code Availability

The data composed of DOIs and associated structured JSON outputs can be found
online at https://doi.org/10.6084/m9.figshare.19719310.v3.[gpt3_data] The
texts for the paragraphs in each paper are excerpted due to copyright
restrictions.

## Author Contributions

A.J., G.C., and K.A.P. supervised the research. K.C. wrote the data collection
infrastructure, performed the data collection, and wrote and applied the
initial gold nanoparticle article classification and information extraction
models. S.G. provided experimental domain knowledge necessary for the template
design. J.D. introduced the GPT-3 sequence-to-sequence information extraction
methodology and prepared the graphic representation of the extraction
template. N.W. co-developed the GPT-3 sequence-to-sequence information
extraction methodology, designed the extraction templates, wrote the code for
interfacing with GPT-3, performed all annotations, performed all GPT-3
experiments, and prepared all results. S.L. provided additional result
validation. All authors contributed to the discussion and writing of the
manuscript.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

This work was funded and intellectually led by the U.S. Department of Energy,
Office of Science, Office of Basic Energy Sciences, Materials Sciences and
Engineering Division under Contract No. DE-AC02-05CH11231 (D2S2 program
KCD2S2). Additional funding used for data set generation via the OpenAI API
was provided by Toyota Research Institute through the Accelerated Materials
Design and Discovery program. This research used resources of the National
Energy Research Scientific Computing Center (NERSC), a U.S. Department of
Energy Office of Science User Facility operated under Contract No. DE-
AC02-05CH11231. This work also used the Extreme Science and Engineering
Discovery Environment (XSEDE) GPU resources, specifically the Bridges-2
supercomputer at the Pittsburgh Supercomputing Center, through allocation TG-
DMR970008S.[xsede]

main


# Knot Dynamics

Louis H. Kauffman,
Department of Mathematics,
851 South Morgan Street,
University of Illinois at Chicago,
Chicago, Illinois 60607-7045,
<EMAIL_ADDRESS>

Abstract The paper studies the dynamics of knots under self-repulsion produced
by artificial charges placed on the curve in space representing the knot.

##  1 Introduction

It is the purpose of this paper to ask questions about the dynamics of knots
under self-repulsion. A knot is arranged in three dimensional space so that it
is coated with a mathematical version of electrical charge so that it evolves
to geometric positions that minimize its energy under constraints that
maintain its total length. Will such evolutions result in the unknotting of
loops that are topologically unknotted? What are the forms of extermal
positions in such evolutions?

We will give a guided tour of computer experiments that can be performed in
the quest for answers to these questions. In the course of our specific
computer exploration we use the knot theory of rational knots and rational
tangles [5] to produce classes of unknots with complex initial configurations
that we call hard unknots and corresponding complex configurations that are
topologically equivalent to simpler knots. We shall see that these hard
unknots and complexified knots give examples that do not reduce in the
experimental space of the computer program. That is, we find unknotted
configurations that will not reduce to simple circular forms under self-
repulsion, and we find complex versions of knots that will not reduce to
simpler forms under the self-repulsion. It is clear to us that the phenomena
that we have discovered depend very little on the details of the computer
program as long as it conforms to our general description of self-repulsion.
Thus, we suggest on the basis of our experiments that sufficiently complex
examples of hard unknots and sufficiently complex examples of complexified
knots will not reduce to global minimal energy states in self-repulsion
environments. In the course of the paper we will make the character of these
examples precise. It is our challenge to other program environments to verify
or disprove our assertions.

To make this challenge quite specific, here is a family of examples. Let

$U_{n}=N([[n+1]]-[[n]])=[(n+1).n]$

in the notation of Section 3 of this paper. $U_{n}$ is a diagram for an
unknot. The example $[11.10]$ is shown in Figure 17 and Figure 18. We
conjecture that any algorithm based only on self-repulsion will be unable to
unknot $U_{n}$ for some natural number value for $n.$ In this example, our
version of KnotPlot is unable to unknot $[11.10].$

More generally, again using the notation of Section 3, we have an infinity of
unknots in the form $N(T-S)$ where $T=(a_{1},a_{2},\cdots,a_{n-1},a_{n})$ is a
rational tangle in continued fraction form, and
$S=(a_{1},a_{2},\cdots,a_{n-1})$ is the truncate of T obtained by removing the
last term of $T.$ These unknotted configuratons provide a wealth of inputs for
knot dynamical experiments. In Section 3 we explain how to create complex
configurations of given knot types by the same method.

Having made this challenge, we point out that in [12] refined methods of
gradient descent provide ways in which many examples that we do not reach with
our methods can be unknotted. And in [9] the BFACF algorithm [8] is used to
strong effect. The challenge is given to all of these more powerful algorithms
as well.

A knot is represented by a (cyclic) sequence $p_{1},p_{2},\cdots,p_{n}$ of
points in three dimensional space $R^{3}.$ In computer graphics the points can
be shown and edges can connect point $p_{i}$ with point $p_{i+1}$ and of
course $p_{n}$ with $p_{1}.$ Another computer graphic option is to enclose the
resulting piecewise linear curve with tube so that the knot appears as a
smooth tube with a chosen diameter. We make use of the computer program
KnotPlot [10] created by Robert Scharein, and we thank him for numerous
conversations about the use of this program, and for modifying the program to
facilitate the use of tangles and rational knots. We shall refer to KnotPlot
when speaking of experiments and examples produced in the course of the paper.

A knot undergoes dynamics of self-repulsion in the computer program by a
recursion involving forces of order $r^{-5}$ ( $r$ is the distance between a
chosen pair of points on the knot) applied to all pairs of non-adjacent points
in the sequence of points that define the knot (see above). The recursion is
controlled so that the knot-type of the space curve does not change from step
to step. We can calcluate the Simon Energy [13]. The Simon Energy is the sum
of $1/|p-q|$ over all pairs of distinct points in the list of points in three
dimensional space that describes the knot. The KnotPlot program minimizes this
energy by self-repulsion forces of order $1/|p-q|^{5}.$ The Simon Energy
corresponds directly to forces of the form $1/|p-q|^{2}.$

One can examine many aspects of the knot dynamics. Knots and links evolve into
special geometric configurations that are not predictable beforehand. One
often finds that unknotted curves will unfold and become simplified into
curves that are nearly flat circles. In some cases the force field cannot undo
unknots. The final configuration can depend upon the starting configuration
and upon degrees of perturbation that are allowed in the process of evolution.
The program we use allows, in addition to the repulsion forces, an attraction
of nearby points via spring-forces (using an analog of Hooke’s Law) [11]. The
balance of the spring and repulsion forces can be used to facilitate
experiments. We will give examples of all these phenomena.

##  2 Moving Down Energy Gradients

We begin with a relatively flat representation of the trefoil knot, a torus
knot of type $(2,3).$ Generally we consider torus knots of type $(a,b)$ where
$a$ and $b$ are relative prime positive integers. Such a knot winds around a
standardly embedded torus in three-space $a$ times around the meridian
direction and $b$ times around the longitude direction. It is a fact that a
torus knot of type $(a,b)$ is topologically equivalent to a torus knot of type
$(b,a).$ We shall write $T(a,b)$ for a torus knot of type $(a,b)$ and so write
that $T(a,b)$ and $T(b,a)$ are topologically equivalent. We shall see this
equivalence emerge in the dynamics.

In Figure 1 we see the initial representation of $T(2,3).$ In Figure 2 we see
the result of the trefoil after it has reached self-repulsion stability. At
this point the trefoil configuration has calculated Simons energy
$E=169.041560.$ The Simons energy is computed by using the fact that the knot
we see is represented by a collection of points in three dimensional Euclidean
space as shown in Figure 3. The Simons energy is the sum of $1/r(p,q)$ ranging
over all pairs of points $(p,q)$ in this configuration where $r(p,q)$ is the
Euclidean distance between the points $p$ and $q.$ The Simon energy
corresponds classically to the force field with exponent $2,$ but it is
convenient to track this energy even when using other force functions.

In Figure 4 we see the initial state for a torus knot of type $T(3,2).$
Topologically, a torus knot of type $T(a,b)$ is equivalent to one of type
$T(b,a),$ but geometrically this is quite a different starting position. In
one case the curve goes around the torus longitudinally twice and meridianally
three times. This is reversed to three and two in the second case. We now
examine how the repulsion dynamics worked on the $T(3,2).$ In Figure 5 we see
the self-repelled $(3,2)$ torus knot in what appears to be a stable position.
Experimentally this configuration is stable when using the undamped program in
KnotPlot. Undamped means that the springs that are available between
successive nodes in the model are not operation (in the sense that the spring
constant is high and the springs are hence fully contracted). The energy level
of this configuration is $174.59304$, considerably higher than the minimum we
found for the $(2,3)$ knot. Now we have the opportunity to perturb this
configuration by using the program in its undamped form. Then the springs are
free to contract and expand, letting the energy move into kinetic energy of
motion for the configuration. In Figures 6 7 8 9 we see that indeed the
undamped evolution results in an oscillation that swings arcs of the knot
around and lets it fall into the lower energy configuration of the $(2,3)$
trefoil knot.

Figures 10 11 12 13 14 15 illustrate the descent of a $(5,2)$ torus knot. The
resulting energy minimum is $221.466258.$ Note that the internal twist in the
knot collapes in the course of this evolution. The same collapse will be
observed for $(n,2)$ torus knots with $n\geq 5.$ If one uses the undamped
descent for a long time, then for the $(5,2)$ the resulting configuration does
not change when the undamped evolution is made available. If one runs the
damped evolution for a short time, then the result will be similar to our
evolution with the trefoil knot and an exchange of kinetic energy takes the
knot to the same configuration as the descent of the $(2,5)$ torus knot. We
illustrate the final descent configuration for a $(2,5)$ torus knot in Figure
16.

In Figure 17 we illustrate an initial position for an unknot of type $[11.10]$
where we will explain this notation below. The Figure 18 shows the result of
the undamped evolution of this knot. It has not become unknotted and no
application of the undamped program will unknot it at any point in the
process. This is an eperimental example of an unknnotted curve that fails to
become unknotted in the force field of the Knotplot program. We conjecture
that this phenomenon is not specific to this program but rather that some
unknots of the form $[(n+1).n]$ will not be able to evolve to simple circles
in any forcefield of this kind. The reader is advised to experiment with these
examples. We explain the corresponding tangle theory in the next section.

The knot shown in Figure 19 is obtained by the same techniques as the previous
two figures. It is of type $[15.10]$ and is equivalent to a figure eight knot
as we will explain in the next section.It will not reduce any further in the
KnotPlot environment either via undamped or damped evolution. In fact, as we
will see in then next section, we have an infinite collection of figure eight
knots in the forms $[(n+5).n]$ and for $n$ greater than $10$ we have the
empirical result that they all have local energy minima , increasing with the
value of $n$ in the KnotPlot environment. We conjecture that this phenomenon
is qualitatively the same in any force field environment for knot repulsion.

---
Figure 1: Initial Trefoil
---
Figure 2: Self Repelled Trefoil
---
Figure 3: Beaded Trefoil
---
Figure 4: (3,2) Torus Knot
---
Figure 5: Self Repelled (3,2) Torus Knot
---
Figure 6: Spring Dynamics Destabilizes the (3,2).
---
Figure 7: Spring Dynamics Destabilizes the (3,2).
---
Figure 8: Toward a New Minimum.
---
Figure 9: Final Minimum Equivalent to the (2,3) Trefoil Minimum.
---
Figure 10: Initial condition of $(5,2)$ torus knot.
---
Figure 11: Descending $(5,2)$ torus knot.
---
Figure 12: Descending $(5,2)$ torus knot.
---
Figure 13: Descending $(5,2)$ torus knot.
---
Figure 14: Descending $(5,2)$ torus knot.
---
Figure 15: Final Descent of $(5,2)$ torus knot.
---
Figure 16: Minimum for the (2,5) torus knot.

---
Figure 17: Starting Position for the unknot $[11.10]$.
---
Figure 18: End Position for the unknot $[11.10]$ in Local Energy Minimization
- A high energy version of the unknot.
---
Figure 19: End Position for the knot $[15.10]$ in Local Energy Minimization -
A high energy version of the figure eight knot.

##  3 Hard Knots and Collapsing Tangles

Before we begin the details of this section, the reader should view Figure 26.
There is illustrated a diagram and a sequence of steps that show that the
diagram is unknotted. This is the type of unknot that we have discussed in the
previous section. In order to systematically construct configurations of this
type, that are complex and yet unknotted, we develop an algebraic language of
tangles in the present section of the paper. For complete information about
all aspects of this discussion we recommend that the reader consult the paper
[5].

In this section we give background in terms of a theory of tangles for
constructing the unknots and knots in collapsible states. We begin with the
definition of a tangle and the notions of combination of tangles (addition,
star product, numerator and denominator) and how this leads to the tangle
fraction, the notion of rational tangles and rational knots and how these
constructions are related to computer experiments.The notion of a tangle was
introduced in 1967 by Conway [1] in his work on enumerating and classifying
knots and links.

A $2$-tangle is a proper embedding of two unoriented arcs and a finite number
of circles in a $3$-ball $B^{3},$ so that the four endpoints lie in the
boundary of $B^{3}$. A tangle diagram is a regular projection of the tangle on
an equatorial disc of $B^{3}.$ A rational tangle is a special case of a
$2$-tangle obtained by applying consecutive twists on neighbouring endpoints
of two trivial arcs. Such a pair of arcs comprise the $[0]$ or $[\infty]$
tangles as shown in Figure 21, depending on their position in the plane. We
shall say that the rational tangle is in twist form when it is obtained by
such successive twists of adjacent strands. For example see Figure 20 where we
show a tangle in the center of the figure that is obtained by first starting
with a horizontal 3-twist ($[3]$ in Figure 21 and then twisting two vertical
strands negatively twice and then twisting two horizontal strands positively
twice. Tangles obtained recursively from the untwisted tangles $[0]$ and
$[\infty]$ of Figure 21 are called rational tangles.

Note that in Figure 21 we have indicated horizontal twists by bracketed
integers $[n]$ and vertical twists by reciprocal integers $1/[n],$ including
the convention that $1/[0]=[\infty].$ In Figure 22 we show tangle operations
$T+S$, $T\star S$, and $T^{rot}.$ These are called addition, star product and
rotation of tangles. The rotation operation has the effect of turning the
tangle by ninety degrees in the plane. Note that twisting horizontal strand
$n$ times is the same as forming $T+[n]$ and twisting vertical strands $n$
times is the same as forming $T\star[n]$. Thus we can describe the tangle in
Figure 20 as ($[3]\star[-2])+[2]$.

It turns out that one can associate to any tangle $T$ a fraction $F(T)$ with
values in the rational numbers plus the formal value $\infty,$ and that this
fraction satisfies the following rules with respect to addition, star product
and rotation. Here $T^{\star}$ denote the mirror image tangle, obtained by
switching all the crossings of $T.$ Once we assume that $F([0])=0,F([1])=1,$
these rules determine the fractions of rational tangles.

  1. 1.

$F(S+T)=F(S)+F(T)$

  2. 2.

$F(T\star S)=\frac{1}{\frac{1}{F(T)}+\frac{1}{F(S)}}$

  3. 3.

$F(T^{rot})=-1/F(T)$

  4. 4.

$F(T^{\star})=-F(T)$

The second property of the fraction for the star product is a conseqence of
first property for addition, using the third property that tells us that the
fraction of a rotate is the negative reciprocal of the fraction of the
original tangle. Note that it follows from these rules that $F([n])=n$ and
that $F(1/[n])=1/n$ and that

$F(T\star(1/[n]))=1/(n+1/F(T)).$

This is the formula for the fraction of a tangle with an extra vertical twist
$1/[n]$ at the bottom. From this formula we see that the fraction of a
rational tangle will be represented by a continued fraction. For example,

$F([a]+([c]\star(1/[b])))=a+1/(b+1/c).$

For this reason it is useful to denote rational tangles by just writing a
corresponding continued fraction.

---
Figure 20: Tangles - Numerators and Denominators
---
Figure 21: Elementary Rational Tangles
---
Figure 22: Addition, Star Product and Rotation of Tangles
---
Figure 23: Finding the Fraction of a Rational Tangle
---
Figure 24: Rotational Isotopy
---
Figure 25: Simplifying a Tangle Sum
---
Figure 26: $N((1,2,3)-(1,2))$ is Unknotted.
---
Figure 27: $N(a,b,c)=N(c,b,a)$
---
Figure 28: $N((a,b,c)+(d,e,f))=N(c,b,a+d,e,f)$
---
Figure 29: Second example of the numerator of a sum of tangles.

---
Figure 30: $N(7,7,7,7)$ – Initial State.
---
Figure 31: $N(7,7,7,7)$ – Final State.
---
Figure 32: Last Knot – Initial State.
---
Figure 33: Last Knot – Final State.

We will illustrate this with an example. In Figure 23 we illustrate these
properties by starting with the tangle $T$ from Figure 20 and noting that $T$
is a sum of the form $T=S+[2]$ where $S=[3]\star[-2].$ We could directly use
the rules above to compute as follows.

$F(S)=F([3]\star F(1/[-2])=1/(1/F([3])+F([-2]))$ $=1/(1/3+-2))=3/(1-6)=-3/5.$
$F(T)=F(S+[2])=F([S])+2=-3/5+2=7/5.$

In the Figure 23 we denote $x=F(S)$ and use property $3.$ above to rotate the
tangle and then use property $1.$ to complete the calculation of $F(S)$
without using property $2.$ In the rest of the figure we note that $7/5$ has a
positive continued fraction expansion as
$7/5=1+2/7=1+1/(7/2)=1+1/(2+1/2)=[1,2,2].$ The last symbol, $[1,2,2]$ is the
symbol for this continued fraction. Thus

$[a,b,c]=a+1/(b+1/c)$

and more generally

$[a_{1},a_{2},a_{3},a_{4},\cdots,a_{n}]=a_{1}+1/(a_{2}+1/(a_{3}+1/(a_{4}+\cdots+1/a_{n})))).$

Thus we shall from now on use the continued fraction written numerically to
indicate the corresponding rational tangle. Thus $(1,2,2)$ will be our
notation for the tangle diagram at the bottom of Figure 23. Note that
topologically this tangle is equivalent to $(1,3,-2).$ These symbols represent
the same numerical fraction and they each represent specific diagrams for
rational tangles. The Figure 23 gives the reader enough information to see
that these two tangles are topologically equivalent. That is, the reader can
take as an exercise to deform the tangle $(1,3,-2)$ to the tangle $(1,2,2)$ by
an isotopy that does not move any arcs over the endpoints of the tangle, while
leaving the endpoints fixed.

John Horton Conway proved that two rational tangles are topolgically
equivalent if and only if they have the same fraction. See [1, 5, 6, 7] for
statements and proofs of the Conway Theorem. The key to our approach to
proving the Conway Theorem, in these cited papers, is the insight that any
continued fraction tangle with a mixture of signs such as $(1,3,-2)$ is not an
alternating weave, and can be simplified by standard isotopies (swinging arcs
that go over or under consecutive crossings) until it is alternating. The
rational form of the tangle is retained under such isotopies, and the final
alternating form corresponds to an entirely positive (or entirely negative)
sequence such as $(1,2,2).$ The only positive or only negative sequence is
uniquely obtained from the fraction itself by Euclid’s algorithm. We give no
more details here, but refer to our papers.

Note that $a+1/(b+1/c)=(a,b,c).$ Thus $(0,b,c)=1/(b+1/c)=1/(b,c).$ These
sequential symbols are useful for handling rational tangles and continued
fractions. Note also that when we draw $(a,b,c)$ the twists occur from right
to left. Thus in Figure 23 we illustrate $(2,-2,3)$ and the $2$ occurs on the
right in the diagram.

In Figure 24 we illustrate a useful fact about about tangles. If $T$ is a
tangle and we apply rotation to it twice it will be rotated by 180 degrees. It
turns out that for rational tangles, one can pick them up, rotate them by 180
degrees and they are topologically unchanged. The figure illustrates one case
of the internal topological rotations that accomplish this fact.

The next operation we need is the numerator construction $N(T)$ that converts
a tangle to a knot of link by connect the top strands to one another and the
bottom strands to one another. See Figure 20 for an illustration of the
numerator operation and its so-called corresponding denominator operation. A
theorem of Schubert (See [LKS]) gives a classification of the rational knots
obtained as numerators of rational tangles. We do not need the full Schubert
Theorem here but refer to [7] for the reader who wants more information about
it.

In Figure 25 we illustrate the formula $N(T+[n]+S+[-n])=N(T+S).$ Actually this
is simply an identity for tangles, but it is often occurring inside a
numerator closure. Note that algebraically this is just the cancellation of
$[n]$ with $[-n]$ but topologically it can be interpreted as the result of a
rotation that involves the intermediate tangle $S.$ Such rotations occur in
manipulating rope models and in the motions of curves in the force fields we
have considered in the first section of the paper.

Now we are in position to look at Figure 26 again. Note that the diagram
indicated is $N((1,2,3)-(1,2))=N([1]+(0,2,3)+[-1]+(0,-2))$ and so the $[1]$
and $[-1]$ cancel is in the previous paragraph. Again one can see this by an
internal rotation in the numerator closure. The rest of the figure illustrates
how this configuration is unknotted.

We can note that if $T=[a_{1},a_{2},\cdots,a_{n}]$ and
$T^{!}=[a_{n},a_{n-1},\cdots,a_{1}],$ the tangle obtained by reversing all the
terms in $T,$ then $N(T)$ and $N(T^{!})$ are topologically equivalent knots.
The Figure 27 will give the reader an appreciation of this fact. In that
figure we illustrate how $N(2,3,4)=N(4,3,2)$ by representing each of these
tangles in a partially braided form. The reader can take as an exercise to see
that any continued fraction template for a rational tangle can be put into
such a braided form.

Figure 28 illustrates $N(T+S)$ for $T=(2,3,4)$ and $S=(3,2,2).$ We have the
basic fact that the numerator of the sum of two rational tangles $T$ and $S$
is topologically equivalent to the numerator of a rational tangle $T\square S$
that is constructed from the two of them. Given $T=[a_{1},a_{2},\cdots,a_{n}]$
and $S=[b_{1},b_{2},\cdots,b_{n-1},b_{n}],$ let $T\square S$ be defined by the
formula

$T\square
S=(a_{n},a_{n-1},\cdots,a_{2},a_{1}+b_{1},b_{2},\cdots,b_{n-1},b_{n}).$

Then $N(T+S)$ is topologically equivalent to $N(T\square S).$ In order to see
this topological equivalence the Figure shows how to rotate and isotope
$N(T+S)$ so that it is clearly equivalent to $N((T\square S)^{!}).$ But we
know that $N(T^{!})=N(T)$ for any $T.$ So this gives the desired result. Thus
in Figure 28 we see that

$N((2,3,4)+(3,2,2))=N(4,3,2+3,2,2)=N(4,3,5,2,2).$

The rotation that we see in $N(T+S)$ in Figure 28 is the key to producing many
diagrams of the unknot that can be difficult for the dynamics of the program
to undo. The basic result is this: Let $T=[a_{1},a_{2},\cdots,a_{n}]$ and
$S=[a_{1},a_{2},\cdots,a_{n-1}].$ Then $N(T-S)$ is topologically equivalent to
the unknot. Figure 26 illustrates how this works. The tangle $S$ has just one
less term than the tangle $T$ and is obtained from it by removing the last
term of $T.$ We call $S$ the truncate of $T.$ In the figure the reader should
be able to see that the $a_{1}$ from $T$ and the $-a_{1}$ from $-S$ flank a
tangle and allow a rotation that cancels these terms and then brings $a_{2}$
and $-a_{2}$ into cancelling position. This continues recursively and allows
the knot to unwind and become unknotted. To see this result in tangle terms
the following identity is of use:

$(\cdots a,b,0,c,d,\cdots)=(\cdots a,b+c,d,\cdots)$

where this equality can be read as topological equivalence of tangles, or as
identity of continued fractions. For example, in fractions we have

$(a,b,0,c,d)=a+1/(b+1/(0+1/(c+1/d))))$ $=a+1/(b+1/(1/(c+1/d))))$
$=a+1/((b+c)+1/d)$ $=(a,b+c,d).$

The unknotting is then mirrored in a bit of algebra. For example

$N((a,b,c,d)-(a,b,c))$ $=N(d,c,b,a-a,-b,-c)$ $=N(d,c,b,0,-b,-c)$
$=N(d,c,b-b,-c)$ $=N(d,c,0,-c)$ $=N(d,c-c)$ $=N(d,0)$

and $N(d,0)$ is unknotted.

By the same token, we can arrange tangle diagrams that collapse to certain
knots. For example,

$N((1,1,1,c,b,a)-(1,1,1))$ $=N(a,b,c,1,1,1-1,-1,-1)$ $=N(a,b,c,1,1,0,-1,-1)$
$=N(a,b,c,0)$ $=N(a,b)$

This means that we can give the dynamics program tangled versions of knots and
unknots and perform experiments to see if the program will undo the unknots
and collapse the complexified knots to simpler forms.

###  3.1 Collapse Examples

Notation. We make the following definitions. Let $[[n]]=(1,1,1,\cdots,1)$ with
$n$ $1$’s in the continued fraction sequence. Let $[n.m]=N([[n]]-[[m]]).$ This
is the notation we have used in the first section of the paper. Thus Figure 17
and Figure 18 give images for $[11.10],$ a configuration that we know is
unknotted from the above construction. These figures show that the KnotPlot
program takes $[11.10]$ to a stable state that is still entangled. The second
instance is Figure 19 where we show how $[15.10]$ also has a complex stable
state. By our discussion below we will see that for $n>m$,
$[n.m]=N([[n-m-1]]).$ Thus $[15.10]=N([[6]])$ and this is a Figure Eight
knot.The upshot of these constructions is that we can give infinitely many
examples of configurations that are either knotted or unknotted but have
complex stable states in the force field algorithm of KnotPlot.

To prove that $[n.m]=N([[n-m-1]]),$ consider an example:

$[7.3]=N((1,1,1,1,1,1,1)+(-1,-1,-1))$ $=N(1,1,1,1,1,1,1-1,-1,-1)$
$=N(1,1,1,1,0)=N(1,1,1)$ $=N([[7-3-1]]).$

Thus the proof of this fact follows from the work we have already done, plus
the use of this compressed notation.

We will also write $([[n]],a,b,c)=(1,1,1,\cdots 1,a,b,c)$ so that the $[[n]]$
can indicate $n$ consecutive $1$’s in the tangle. Then we have the basic
result that

$K_{n}(a,b,c,d)=N([[n+1]],a,b,c,d)-([[n]]))$

is isotopic to

$N(a,b,c,d)$

where $(a,b,c,d)$ can be replaced by any continued fraction tangle. This
result follows by the same reasoning as we have outlined above, and it
provides a large collection of starting configurations for testing a self-
repulsion program.

Our last example illustrates all the principles in this section. The basic
knot we work with is the rational knot $N(7,7,7,7)$ illustrated in its initial
form in Figure 30 and it its final fully repelled form in Figure 31. In Figure
32 we show the initial form of $K_{11}(7,7,7,7)=N(([[12]],7,7,7,7)-([[11]]).)$
In Figure 33 we show the final self-repelled form that occurs from this
beginning. We see that in this case $K_{11}(7,7,7,7)$ does not reduce to the
minimal form of $N(7,7,7,7)$ in the KnotPlot environment. It is our conjecture
that no environment of this type will fully reduce all of the configurations
$K_{n}(7,7,7,7),$ and we challenge users or designers of other programs to
experiment with this family of configurations of the knot $N(7,7,7,7).$ There
is, of course, nothing special about the choices we have made here. The
phenomenon of non-collapse will occur for sufficiently large complexifications
of any knot or unknot.

## References

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