The paper was printed toward the rear of the Journal of Combinatorics, spring 1978. Andret read it late on a Monday afternoon. He stared at the poorly printed page in the small lounge where the department shelved its literature. A steeple bell chimed the hour; but he hardly heard. Someone entered the room, poured a cup of coffee, and left. Andret turned back the page and reread the paragraph.
Erdős had cited a method he’d developed to discern the probability that one of his opponents had changed direction in the game. Andret closed his eyes for a moment, then went back and read the paragraph again.
This was it. This was the way to breach the Abendroth.
He shut the journal, set it back in its place on the shelf, and looked around, as though he’d been caught at something.
Another Roof, Another Proof
NOW HE BEGAN a search. The instructive theoretical instance was the prize he sought. When he arrived at the office, he would clear the desk of yesterday’s notes, then spend the morning in thought — it might take three or four hours to assemble a single figure in his mind. In the afternoons, he would sketch, committing what he’d composed to memory. At first he tried to conceive of a figure that might invalidate the approach he’d picked up from Erdős; but then, as such figures one by one proved assailable, he began to work on examples that might support it. At home on his bed table, he kept a pen and paper, in case something came to him at night.
For months, he pushed on. The proof might require another three years of work, maybe four — but so what? To solve two great problems in a lifetime would bring him to the pinnacle of his profession. He asked one of the secretaries to buy a half-dozen cartons of laboratory notebooks. When they arrived, he numbered their bindings, then filled the pages with variations on certain theoretical shapes—3-manifolds parsed into every sort of Heegaard Splitting and torus decomposition that he could imagine.
Then he began, slowly and at first by negation, to bridge the moat around the problem. It was tedious work, but the moat needed to be crossed before he could scale the walls. In a day he could fill an entire pad with drawings. As he worked, he felt warmed through his body, down to his hands and feet, as though it were vigorous physical exercise he was performing and not a motionless feat of endurance, sitting still at his desk for hours. He kept a window open for air. Sometimes he grew breathless nonetheless, as first he sensed himself moving to the edge of the moat, then starting across it. On he went. Into numbered file boxes he laid numbered notebooks, which in turn he stacked in numbered sequence along the bare walls of the office. He didn’t need his own drawings in order to think, as did some of his lesser colleagues; but he knew that he would need them later for reference — a year from now, five years from now — when he went over the wall into the castle.
—
IN OLGA PETRINOVA’S basement apartment, the radiator was set so low that she kept a sweater on indoors. But underneath the sweater was always a dress, pleasingly stretched, and underneath the dress a queerly sewn, Bolshevik-style undergarment that he grew to crave. Between the knees and breasts it was a thick gray wool, but around the hems ran lacy sinuations of black silk that might have come from a shop in Paris.
The revived state of his work made him ravenous.
Afternoons, he visited. At the door, which was down a flight of stairs, she would greet him with her hands on her hips, her breasts thrust forward through the sweater in a way that was both flirtatious and accusatory. Was this only in his mind? At the small thrift-store table by the refrigerator they would drink a pair of bourbons, then slide the chairs across the room to wait for the sun. Her hands smelled of fennel, and there was something not quite real about the color of her hair, but her bony beauty never failed to catch him. By 2:30 or so, when the sun and the bourbon had warmed her enough, she would narrow her eyes.
In bed she liked to talk, a little before and a little after, both of which were courtesies that the drink allowed him. Before she would let him touch her, she would converse with him solemnly for a few minutes, about Soviet politics or academic mathematics, the way other women might talk about the roses or the house. She seemed to regard these moments as a test of her character. As she talked, Andret would make gentle, two-fingered tugs all the way around the hem of her dress to expose the lacy parts of her undersuit, like a child pulling candles from the rim of a birthday cake. Then he would begin kissing the frills. This she found beguiling. During sex she would quiet, moving suddenly on top of him like a lion over its prey. Her eyes stayed wide. Andret liked to keep his own closed; but whenever he opened them, there she would be, staring down at him, her black pupils gyroscopically inert. Again: leonine. He couldn’t help thinking that her gaze, even as she bent over him and strained her shoulders like a collared beast, was in fact an indictment.
The act itself was fervent. Like a brisk tennis game or a summer track meet, something performed in daylight between competitors. The cheap mattress bounced. She liked to do it more than once, and he was usually able to comply. Bourbon was his gasoline. Between sessions, he poured it at the counter while she lay panting on the sheets. Sweat burnished her body. The lean neck. The surprisingly full breasts. He would down another glass and return. The competition would continue in the relentless searching of her eyes, which never rested. Sometimes there was even a third time, in which those eyes, even as they sharpened with arousal, then fogged with it, continued their accusation: he was a cad; he was really just using her; he cared little for anybody.
He was aware, of course, that all of this was true.
—
ANNABELLE DETMEYER, ON the other hand, would greet him at the servants’ door of the Detmeyer estate, humid eyed and already wrapped in some easy-to-remove vestment. In the kitchen a meal would be waiting. Some sumptuous-smelling casserole warming in an iron pot beside a pair of parsley-garnished bowls. Heated bread beneath a square of linen. The Detmeyers had two girls, five and nine, but the house had a half-dozen bedrooms on the second floor alone and an entire top story above that. Behind the property ran a tract of forest preserve. He used the preserve for cover, entering it behind a parking lot in town and emerging like a spy at its near end, across from the tamarack that hid the back door of the Detmeyer garage. Annabelle preferred his visits in the morning, just after the girls had left for school. Except for the few feet of lawn between the woods and the first branches of the tamarack, it was an entirely cloaked journey; but he wore an overcoat anyway, and a pair of dark glasses. Her husband traveled nearly every week.
Bookshelves lined the walls of the house, not only in the library and study but in the living room and kitchen and all the bathrooms.To Andret’s mind, this gave the whole downstairs the appearance of a rambling, geometric painting, in which the colored spines of thousands of books formed unintentional refuges of color — predominantly green here, predominantly maroon there — the way random patterns tended to do. In the mathematics section of the study, in fact — Yevgeny Detmeyer taught quantitative economics — there were treatises on exactly such patterns. Andret amused himself with the titles.