Endless Knots
The clues are presented as infinitely scrolling and the given integers p and q are coprime. This along with the title “Endless Knots” hints towards the main twist (pun intended) of the puzzle. Each list of clues should be associated with a torus knot defined by the (p,q) values.
A torus knot is usually a 3D shape. However in this case each set of clues forms what is arguably a diagramless crossword where the grid is a 2D representation of the torus knot. Concatenated answers to the clues weave through each other around the grid and therefore this string of letters has no beginning nor end (hence the presentation). The Example set provides a playground to experiment with how to fill a grid on a small scale. Two clues are bolded. Reading them tells us that in order to complete a grid, we need to FILL BLANKS with one letter per blank in the torus knot diagram.
A sketch of the filling pattern from an arbitrary starting point is shown below using the torus knot grid from the first subpuzzle. We can read answers to clues from the cells this line passes through. The outside and inside edges always contain unchecked letters and everything in between contains checked letters. Since all word lengths are given, these properties may be used to logically deduce positions of letters in a grid.
There are two things to extract from each grid. One is an unchecked letter where the “? (1)” clue appears in each list. The other is a word or phrase. The extraction gimmick for the latter is clued by the title of each subpuzzle. In all but Example and Edged Case this gimmick is also required to correctly fill the grid.
The single letters together yield an answer parsed as READ AS SI. But what do we read as an SI metric? Indexing (p,q) into the extracted word or phrase and parsing in column major order yields TEN NEG TWELVE. This is a metric SI prefix which is the final answer: PICO.
A table summarizing the extraction and fill gimmicks is below:
| Name | p | q | Fill and extraction gimmick | Answer | Index p | Index q | “? (1)” clue(s) |
|---|---|---|---|---|---|---|---|
| Edged Case | 7 | 11 | Read letters on the outside or inside edge (depending on how the grid was constructed). | INFINITESET | T | T | R |
| Twisted Trio | 5 | 7 | There are enough clues to fill the grid thrice. Take the overlapping letters from each subsequent pair of grid fills. | SPIDERWEB | E | W | EAD |
| Missing Halves | 6 | 11 | Three of the clues give answers that don’t fit in the blanks or the enumeration. The enumeration is for only half the answer. Take the halves that don’t fit and read clockwise around the filled grid. | HERRINGBONE | N | E | A |
| desreveR | 7 | 12 | Two of the answers must be filled in backwards. E.g. if all clues are being filled clockwise, these two must be counter-clockwise. | NUTRIENTCYCLE | N | L | S |
| Double Trouble | 7 | 9 | Five cells must be filled with two letters to make the rest fit. Read the bigrams. | INTERWEAVE | E | V | S |
| Differing Opinions | 9 | 11 | Letters in eleven of the cells conflict with their woven overlaps. Take their difference and A1Z26. | DISENTANGLE | G | E | I |
Appendix
Here are some notes on generating the grids.
In the mathematical sense a torus knot of (p,q) is equivalent to one of (q,p). Depending on where one looks up the parameterization to draw the knots these two values are sometimes inconsistently labeled. This puzzle uses the notation that appears on the wikipedia page and also through querying Wolfram Alpha. Here is a simple Python script to generate a torus knot diagram.
The filled torus knot diagrams, completed lists of clues, and any further subpuzzle extraction details are given below. Single letter extraction is marked in blue. Clues relevant to the subpuzzle extraction are marked in red. All grids here are filled clockwise.
Example (p=5;q=4)
Boundaries between words are marked with a bold line.
| Answer | Length | Clue |
|---|---|---|
| LEG | 3 | A lower body appendage, you have two of them |
| LAW | 3 | ___ and Order |
| KITE | 4 | Something you can fly on a windy day |
| BEAN | 4 | It could sprout into a stalk |
| FILL | 4 | When you eat well you might eat your ____ |
| BLANKS | 6 | Training bullets |
Edged Case (p=7;q=11)
Answer is extracted from the outside edge. However it is also possible to fill this knot such that the final answer appears on the inside edge. There is no way to mark where to start reading the answer, however only one starting position gives an answer that is vaguely thematic to the puzzle.
| Answer | Length | Clue |
|---|---|---|
| ONESIE | 6 | Infant garment |
| ETC | 3 | List ender, abbr. |
| ON | 2 | Off's opposite |
| TEA | 3 | Hot leaf juice |
| SPADE | 5 | Gardening tool |
| SCION | 5 | Descendant or heir |
| URGE | 4 | Craving |
| R | 1 | ? |
| TAINT | 5 | Sully |
| BUMP | 4 | This slows down cars on the road |
| SEAT | 4 | An elected position |
| CHIPS | 5 | Potato-based snacks, in both the US and UK |
| RIO | 3 | River in Spain or Portugal |
| ANA | 3 | Former World No. 1 Ivanovic |
| OTAKU | 5 | Someone who likes anime and/or manga |
| SOW | 3 | Female pig |
| EASTER | 6 | Egg or island preceder |
| PUS | 3 | Fluid in infected tissue |
| FIAT | 4 | Unbacked currency |
| CASA | 4 | House in Spanish |
| BARS | 4 | Places to drink or measures in music |
| WEEP | 4 | Willows might do this sadly |
| REIN | 4 | Keep under control |
| EFFORT | 6 | Conscious exertion of power |
| LATTE | 5 | Espresso and steamed milk |
| EROS | 4 | Greek god of love |
| DEPTH | 5 | The fathom measures this |
Twisted Trio (p=5;q=7)
Unlike the other subpuzzles, these three grids must have a “beginning” and “end” to each fill. The boundary between the grids is marked by the bold line. The first grid starts with PIER and ends with ILL, second is PIT to RULER, third is ADHD to SHY. This also allows the grid fills to be overlaid on top of each other. Subsequent overlapping letters give answer substrings. The leftmost grid has the word ONE as part of the fill. Using this grid first we get a reading order for both the subpuzzle answer and the single letter extracts.
| Answer | Length | Clue |
|---|---|---|
| PIER | 4 | ____ 39, in San Francisco |
| NOPE | 4 | A red negation card in Exploding Kittens |
| REBEL | 5 | Someone who resists authority |
| SUIT | 4 | Hearts or diamonds, for example |
| ONE | 3 | In set theory, this can be defined as the set containing the empty set |
| NEWT | 4 | An eye of this is favored by witches for potions |
| BIAS | 4 | Unfairly influence |
| PETRA | 5 | Nabataean capital |
| E | 1 | ? |
| POWER | 5 | Work over time |
| ILL | 3 | Sick |
| PIT | 3 | Hole or cavity |
| DENT | 4 | Light damage on a car |
| GORY | 4 | Bloody |
| MESH | 4 | Material in screen door |
| IDLE | 4 | Inactive |
| RIGA | 4 | Latvian port city |
| XYLO | 4 | Prefix meaning "wood" |
| SPUD | 4 | Informal term for potato |
| DIRTY | 5 | Filthy |
| A | 1 | ? |
| RULER | 5 | Straightedge with marks |
| ADHD | 4 | Neurodevelopmental condition, abbr. |
| EYES | 4 | Visual organs |
| DEBT | 4 | Opposite of credit |
| HEN | 3 | Rooster's mate |
| D | 1 | ? |
| AVERT | 5 | Stave off a crisis |
| SWAB | 4 | Tool in COVID test kit |
| SPEAK | 5 | To utter words |
| ADORE | 5 | Admire and love |
| AWED | 4 | Filled with wonder |
| SHY | 3 | What someone might be around their crush |
Missing Halves (p=6;q=11)
Take the second halves of the clues FAT|HER, COVE|RING, and WISH|BONE and read clockwise. Again only one starting position gives an answer that makes sense.
| Answer | Length | Clue |
|---|---|---|
| SMACK | 5 | Loud kissing sound |
| DODGE | 5 | Evade |
| IRON | 4 | Magnetic metal |
| HARP | 4 | Instrument associated with angels |
| AFAR | 4 | At a distance |
| AMINO | 5 | Type of acid that makes up proteins |
| DORA | 4 | Child explorer with Boots |
| EASY | 4 | Lowest difficulty level, typically |
| NEON | 4 | Lights found in a bar |
| PARA | 4 | Prefix indicating a functional group directly opposite on a benzene ring |
| ROMAN | 5 | French novel |
| IVORY | 5 | Ebony's partner |
| WISHBONE | 4 | This is broken in half at Thanksgiving |
| AHOY | 4 | Originally suggested telephone greeting |
| YARN | 4 | Fiction, if you spin it maybe |
| TAME | 4 | Not wild |
| COVERING | 4 | An awning for example |
| DAWN | 4 | Ash's Sinnoh companion |
| RYAN | 4 | American Idol host Seacrest |
| A | 1 | ? |
| NYE | 3 | The science guy |
| FATHER | 3 | Male parent, formally |
desreveR (p=7;q=12)
| Answer | Length | Clue |
|---|---|---|
| MONEY | 5 | This can be used to exchange for goods and services |
| SWAP | 4 | To exchange |
| LYRE | 4 | Harp ancestor |
| HIND | 4 | Rear (of a body part) |
| OUTCAST | 7 | Pariah |
| TOTE | 4 | Type of bag |
| S | 1 | ? |
| TORPID | 6 | Lethargic, such as during hibernation |
| ACHE | 4 | Pain in head or back |
| LIEU | 4 | Stead |
| ORBIT | 5 | Equivalence class of group action |
| ROAM | 4 | Leave a phone's service area |
| SEWN | 4 | Made with thread and needle |
| MICRO | 5 | Prefix represented by mu |
| VELDT | 5 | African grassland |
| SONAR | 5 | ASDIC was an early version of this |
| TRUTH | 5 | The alternative to dare |
| YEAR | 4 | 2023, for example |
| CYCLE | 5 | A periodic sequence |
| SILVER | 6 | Metal commonly associated with runners-up |
| NUTRIENT | 8 | Protein or vitamin |
| META | 4 | Puzzle type that might show up at the end of a round |
| AMP | 3 | Unit of electric current |
| DYE | 3 | Food coloring |
| VOLVO | 5 | Car brand which means "I roll" in Latin |
| SOREST | 6 | Most in need of a massage |
Double Trouble (p=7;q=9)
The realization that there must be five bigrams may be reached by counting the enumerations of the given clues and the number of blank spaces in the knot diagram.
| Answer | Length | Clue |
|---|---|---|
| RHINO | 5 | Animal typically poached for its horn |
| SEPTA | 5 | Organization that runs public transit in some parts of Pennsylvania |
| PRY | 3 | Be nosy |
| LEAVE | 5 | A break from service |
| UVEA | 4 | Layer where iris is located |
| HUE | 3 | Variety of a color |
| STEP | 4 | Part of a ladder |
| AIRWAY | 6 | The A in ABC, in medicine |
| MADEUP | 4 2 | Like everything inWhose Line Is It Anyway |
| OPINE | 5 | Give one's viewpoint |
| FIR | 3 | Christmas tree, perhaps |
| TIP | 3 | Gratuity to one's server |
| ATLAS | 5 | Book of maps |
| SAVE | 4 | Functionality associated with the floppy disk icon |
| OSMOSIS | 7 | Gradual assimilation of knowledge |
| EARWAX | 6 | You might try to remove this with a Q-tip |
| READ | 4 | Type of receipt |
| S | 1 | ? |
| CLAP | 4 | Sound that requires two hands |
| MASTER | 6 | To achieve proficiency in a skill |
| EXPAT | 5 | Person who resides in a foreign country, for short |
| REVEAL | 6 | Divulge |
Differing Opinions (p=9;q=11)
In this grid, pink letters are always part of words that are read from inside to outside. Orange letters are part of words that read out to in. The pink letters are always bigger than the orange letters and the difference between them reads the answer.
| Answer | Length | Clue |
|---|---|---|
| LYS | 3 | Main tributary of the Scheldt |
| SNOUT | 5 | Pig's nose |
| ANTS | 4 | Nest workers |
| FISH | 4 | This word can be creatively spelled "ghoti" |
| EDIT | 4 | Revise before publication |
| POINT | 5 | Intersection of two distinct lines |
| STAR | 4 | Feature in a film |
| PISS | 4 | To urinate |
| TON | 3 | A metric weight |
| ANODE | 5 | Negative terminal |
| CARD | 4 | Ask for someone's ID |
| EARS | 4 | Aural appendages |
| GURU | 4 | Spiritual teacher |
| SUSHI | 5 | California rolls, for example |
| EXCEL | 5 | Microsoft product which confuses things for dates |
| GPS | 3 | Navigation aid, abbr. |
| NEST | 4 | Avian home |
| FAN | 3 | A big one may be rabid |
| CANDY | 5 | American sweets |
| ONYX | 4 | Banded quartz |
| EPIC | 4 | Long poem whose subject is usually one of legend |
| I | 1 | ? |
| DEN | 3 | Lair |
| AESOP | 5 | Author of fables |
| CITES | 5 | Uses as a source |
| FERN | 4 | Non-flowering plant |
| VOLT | 4 | Unit of electric potential |
| EDDY | 4 | Small swirl |
| OTTO | 4 | Ideal thermodynamic internal combustion cycle |
| BRAND | 5 | Mark with an iron poker |
| TREE | 4 | A connected graph with a unique path between any two points |
| HOPE | 4 | Desire for events to materialize |
| OASIS | 5 | Common mirage target |
| ARSON | 5 | Crime involving fire |
| POTUS | 5 | Acronym for the head of the United States' executive branch |
| DNA | 3 | Molecular basis of inheritance |
| AVE | 3 | Abbreviation seen on some street signs |
| NOIR | 4 | Film genre |
Authors' Notes
OV: I am not a mathematician. I just wanted to make a weird circley diagramless crossword. I had no idea what torus knots were until Herman pointed out that the funny shaped diagram I found online to use in my prototype was one. Learn something new every day I suppose.
In the process of writing this I discovered that the (p,q) notation on the wikipedia page was inconsistent. Of course this was technically explained by an inconspicuous little footnote at the very bottom of the page, but we decided to update the page for clarity anyway. Wikipedia is not exactly a primary source, but we know you will look there anyway.