Knots hold a special place in the hearts of sailors and mathematicians.

That's almost self-explanatory for sailors, who have been tying knots as an essential part of their craft for centuries. They even used them to tell how fast they're going over open water.

Mathematicians' fascination with knots is harder to explain, but comes down to their love of counting. The more complicated the knot, the more the knots strands cross each other, the better. And what better way to make a knot more complicated than by taking two knots, cutting each and then joining their open ends together to make an even bigger, more complicated knot?

They even defined a special rule about the practice to calculate how much more complicated the resulting knot would be. They conjectured that if they took the unknotting number for each knot, which is to say the number of steps it would take to tranform the knot into a simple loop (imaginatively called the "unknot"), and added them together, the result would be the unknotting number of the combined, more complicated knot.

This additivity conjecture worked with just about every combination of knots they could throw at it. Until two mathematicians discovered an example where that rule didn't work. Instead of leading to a more complicated knot, they found a knot that became easier to unknot after being joined. It didn't add up as previous generations of mathematicians had believed it should, and because it didn't, they proved the conjecture about the additivity of unknotting numbers is untrue for all cases.

In the following video, Trefor Bazett guides viewers through basic knot theory in an easy to understand presentation before getting to the remarkable disproof of the additivity conjecture in knot theory by Mark Brittenham and Susan Hermiller, which only took them ten years to work out:

Brittenham and Hermiller's paper is here. Bazett has a second video featuring an extended interview with Brittenham and Hermiller.

The knot they found that broke the rule is the (2,7) Torus Knot, an example of which is the featured illustration for this article. It's also known as the 7₁ knot in a different mathematical knot nomenclature system, but we're not going to get into that topic because why complicate knots any more than needed?

Image credit: Blue 7₁ Knot or (2,7) Torus Knot by Jim.belk on Wikimedia Commons. Public Domain.

Measuring economic activity using astronomical and other scientific tools is a relatively recent development that has come into its own during the 21st century.

One of the neater tools are telescopes and cameras mounted on Earth-orbiting satellites that take pictures of the night-darkened surfaces of the world. NASA's Black Marble project uses those images to tracks how the nighttime lights spread across entire regions change in brightness over time, with fully processed imagery now covering the years from 2014 through 2022.

We haven't gotten too far into the data behind the images yet, but what we see in how nighttime lights have changed the years between 2014 and 2022 appears to correspond with how carbon dioxide emissions have changed in those regions. Nighttime lights in Asia, and especially in China and India have become significantly brighter during a period in which those nation's emissions of carbon dioxide have risen. By contrast, regions that have seen falling levels of CO₂ emissions such as the eastern portion of the U.S. and western Europe have seen the brightness of their nighttime lights grow dimmer.

That introduction brings us to the latest measurements of the concentration of carbon dioxide in the Earth's atmosphere, which is measured at the remote Mauna Loa Observatory. The latest data indicates the pace at which carbon dioxide is being emitted into the Earth's atmosphere is continuing to decline, which indicates the world's economic activity is falling.

The following chart reveals how the trailing twelve month average of the year-over-year change in the rate at which the concentration of carbon dioxide in the Earth's air is changing has changed from January 2000 through July 2026.

The rate at which CO₂ is increasing in the atmosphere is nearing levels where it has bottomed during periods of global recessions, or rather, significant regional recessions, in the 21st century.

References

National Oceanographic and Atmospheric Administration. Earth System Research Laboratory. Mauna Loa Observatory CO2 Data. [Online Data]. Updated 5 August 2026.

Image credit: Eastern Hemisphere Map: Total Radiance Change from 2014 through 2022, NASA Earth Observatory images by Michala Garrison, using data from Li, T., et al. (2026). Public Domain.

As expected, the value of goods exchanged between the U.S. and China increased in June 2026.

The U.S. Census Bureau reports a combined total of $34.7 billion worth of goods were processed through U.S. ports during the month. Both outgoing exports to China and incoming imports from China increased month-over-month. Exports rose 5% from $9.117 billion to $9.571 billion and imports grew 7% from $23.508 billion to $25.150 billion between May and June 2026.

Year-over-year, the combined value of goods traded between the U.S. and China has increased by 22%. Digging into that number, we find some really lopsided results. The increase of U.S. exports to China accounts for just 2% of the year-over-year increase, while China's exports make up the remaining 98%.

Reports indicate technology goods such as semiconductors, computing hardware, and AI technology components led the increase in China's exports to the U.S. in June 2026. In addition, U.S. firms importing lower-value goods for the upcoming Christmas shopping season had a strong incentive to pull their orders forward 1-2 months earlier than normal, seeking to beat the clock on new U.S. tariffs.

The following chart shows how the flow of trade between the U.S. and China has evolved from January 2017 through July 2026.

June 2026 saw the twelve month trailing average value of goods directly traded between the U.S. and China record its second increase after having registered declines in each month from February 2025 through April 2026. The cumulative total loss of direct trade between the U.S. and China with respect to a counterfactual projection of what that trade would have been without the tariff war between the two nations adds up to $187.4 billion through June 2026.

Image credit: Port of Seattle cranes on a sunny October day by Ron Clausen on Wikimedia Commons Creative Commons CC BY-SA 4.0 Attribution-ShareAlike 4.0 International.

Depending on how old they are, U.S. teens are having a very different summer job experience in Summer 2026.

For older teens, Age 18 and 19, the job scene is fairly good. The seasonally adjusted number of teens for this portion of the working teen demographic appears to be fairly steady, if not increasing.

But younger teens, Age 16 and 17, are increasingly finding themselves on the outside of the summer job market. In July 2026, the seasonally adjusted number of younger teens counted as having jobs fell to its lowest level since July 2020. Falling from 1,856,000 in June 2026 to 1,771,000 in July, the decrease continues a long-running downward trend that dates back to January 2023.

In addition, the percentage of younger teens who are employed with respect to the population of 16 and 17-year-olds in the U.S. dropped below 20% for the first time since July 2020 as well. Just 19.7% of Americans Age 16-17 had jobs in July 2026.

The following pair of charts shows the seasonally adjusted employment numbers and employment-to-population percentages for younger teens (Age 16-17), older teens (Age 18-19) and the combined population of working teens (Age 16-19) from January 2021 through July 2026.

July 2026 is also a month in which Americans unexpectedly lost jobs. Among industries that are most likely to employ teens, notable declines were seen in the retail, leisure, and hospitality sectors of the economy.

Reference

U.S. Bureau of Labor Statistics. Labor Force Statistics (Current Population Survey - CPS). [Online Database]. Accessed: 9 August 2026.

Image credit: Crow cabin counselor and camper at best sleepaway camp, Upstate NY by ADKPhoto on Wikimedia Commons. Creative Commons CCO 1.0 Universal Deed.

The S&P 500 (Index: SPX) shot up by nearly 3.6% over it's closing value in the previous week, reaching a new record high closing value of 7,757.46 by the end of the trading week ending on Friday, 7 August 2026.

Strong earnings being reported in the early part of the week provided much of the stock market's upward momentum, with technology and materials stocks providing much of the boost. The positive outlooks being reported by many firms in these sectors contributed to shifting the time horizon of investors forward from either 2026-Q3 (or 2026-Q4) to the more distant future quarter of 2027-Q1.

That change in how far forward investors are looking into the future as they set current day stock prices constitutes a Lévy flight event, which can be seen in the latest update of the dividend futures-based model's alternative futures chart.

On Friday, 7 August 2026, an unexpectedly negative jobs report changed the expected timing of when the Federal Reserve will change short term interest rates in the U.S. The CME Group's FedWatch Tool's outlook changed substantially with the unexpectedly sour employment situation report for July 2026. It now projects the Fed will hike the Federal Funds Rate by a quarter point on 28 October (2026-Q4), six weeks later than it projected a week earlier. It also now sees a greater than 50% chance of another quarter point rate hike on 17 March (2027-Q1).

Here are the other market moving headlines from the week that was:

Monday, 3 August 2026

• Signs and portents for the U.S. economy:
• Chinese state traders make large US soybean purchases
• Oil prices drop after Trump cancels attack on Iran to seek nuclear deal
• OPEC+ agrees September oil hike, completing rollback of voluntary cuts
• US manufacturing activity hits more than four-year high; input prices elevated
• Fed minions say they've got this inflation thing covered:
• Fed's Williams expects inflation to ease, says Fed will act if it doesn't
• FULL TEXT Transcript of Reuters interview with NY Fed President Williams
• Bigger trouble developing in China:
• Factories faced weaker demand, higher costs in July as Iran war grinds on
• China factory growth slows to 4-month low in July, survey shows
• BOJ minions team up with U.S. Treasury to prop up Japan's currency, worry about AI demand making inflation 'sticky':
• Japan, US confirm joint yen-buying intervention, signal more action
• Japan, Bretton Woods 2.0, & The End Of The Carry Era
• How a US-Japan pact to hit yen speculators came together
• Bessent ready to repeat joint yen intervention, urges bigger Fed backstop
• BOJ says global AI demand could have sticky inflationary effect
• Wall Street ended higher as markets flirt with record highs
• Wall Street rallies, Dow closes at record on Iran talks optimism

Tuesday, 4 August 2026

• Signs and portents for the U.S. economy:
• Oil prices settle 5% lower after claims of progress in US-Iran talks
• US oil exports in July fall to lowest level in eight months, data shows
• Fed minion claims they have 'open mind':
• Fed's Paulson keeps 'open mind' on rate policy outlook amid high inflation
• Global economy changes developing in China:
• China's EV export boom is starting to show up in the gasoline market
• BOJ minions leave door to more interventions to prop up yen open, JapanGov minion says inflation not so bad:
• Yen holds most gains as intervention keeps speculators on edge
• Japan may not have intervened in FX market on Monday despite yen's surge, BOJ data suggests
• Japan, US likely to intervene again if yen resumes slide, ex-BOJ official says
• Japan's economy minister offers sanguine view on inflation
• Mixed economic signs developing in Eurozone:
• Euro zone factory output near 4 1/2-year high in July but demand still weak
• Iran war hit euro zone consumption especially hard, ECB says
• Wall Street ended higher as the S&P 500 and Dow notched record closes
• Dow, S&P 500 close at record on AI-linked earnings, Mideast deal hopes
• S&P 500 tags an all-time high and these 10 names have helped fuel the rally

Wednesday, 5 August 2026

• Signs and portents for the U.S. economy:
• OpenAI, Anthropic AI agents implicated in new security breaches
• Oil prices rebound after Houthis say they attacked Saudi tanker
• US service sector maintains strong growth pace in July
• Fed minions say they're chafing at the bit to hike U.S. interest rates some more:
• Fed's Schmid calls for tighter monetary policy to tamp down on 'too high' inflation
• Fed's Schmid says finances around AI buildout merit watching
• Fed's Cook ready to raise rates if inflation doesn't start easing
• Bigger trouble developing in China:
• China services growth hits slowest since September 2024, private PMI shows
• China launches global tax hunt going back decades
• Hong Kong insurers' shares slump on report China to tax offshore insurance income
• BOJ minions thinking about hiking interest rate again, are asked to loan more money to Japan's government:
• BOJ debated mounting price risks even upon hiking rates in June, minutes show
• Nudge from Bessent firms case for BOJ rate hike in September
• Bessent says he is sure BOJ chief Ueda will 'do what is best', NHK reports
• Japan PM Takaichi asked BOJ to buy more bonds when needed, Jiji reports
• ECB minions say AI is adding to Eurozone economy growth:
• ECB says investment shift towards AI helps ease drag of uncertainty on euro zone growth
• Dow closes at record on Mideast optimism; SpaceX, AMD drag Nasdaq
• Wall Street concluded mixed after a softer jobs report

Thursday, 6 August 2026

• Signs and portents for the U.S. economy:
• US labor market stable; worker productivity accelerates in second quarter
• Oil settles up $3 as Iran reviews bill to ban US, Israeli vessels from Hormuz
• One Fed minions not so anxious to hike U.S. interest rates, another wishes they already did:
• Fed's Daly says central bank was right to hold rates steady at July policy meeting
• Fed's Musalem says central bank should have hiked rates at last meeting
• Bigger trouble, stimulus developing in China:
• China's July exports growth likely cooled but still robust
• JapanGov minion says they're looking to tap BOJ minions' investment holdings to pay for tax cut:
• Bank of Japan's ETF holdings may help fund tax cut, LDP executive says
• Wall Street ended lower as investors remain focused on Middle East developments
• Stocks close lower as investors eye Mideast talks, earnings
• More S&P 500 stocks outperform benchmark in strongest breadth in a decade, Balchunas says

Friday, 7 August 2026

• Signs and portents for the U.S. economy:
• US suffers unexpected job losses in July, markets dial back rate hike expectations
• U.S. economy unexpectedly lost 23,000 jobs in July
• Expectations going into the July jobs report: US job growth likely picked up in July
• Brent climbs $1 on uncertainty over end to Iran war
• Fed minions expected to back off rate hike plans after jobs report:
• Market cuts odds of Fed hike after jobs data, but economists still see case for tightening
• NY Fed finds little change in inflation expectations in July
• Fed's Barkin says July jobs data matches recent trend
• Bigger trouble, stimulus developing in China:
• Chinese startup Moonshot's AI model breaks out of testing environment, researchers say
• BOJ minions get unexpected boost in ongoing effort to prop up Japan's currency:
• Japan's yen surges after US jobs data, traders wary of intervention risk
• US euro sale to prop up yen blindsided ECB
• Bigger trouble developing in Eurozone:
• Debt burden grows for German suppliers in embattled auto sector, study shows
• S&P closes at record high as soft jobs report eases rate-hike concerns
• S&P 500 rises as it wraps up banner week with traders seeing bright side of dismal jobs report
• S&P 500 posts weekly gains as Tech, Materials lead sector rally

The Atlanta Fed's GDPNow tool anticipates real GDP growth for the U.S. economy of +5.8%, up from the +5.0% annualized growth it indicated at the end of July 2026.

Image credit: Bing Image Creator. Prompt: "An editorial cartoon of a Wall Street bull who is happy that strong earnings are boosting the S&P 500".