# Prime Knots and Jasmine Flower

To me, prime knots are infinitely beautiful. Knot theory is my favorite branch of mathematics. A mathematical knot is any embedding of a circle into 3-D Euclidean space. Think of it as taking a piece of string, tying and twisting it up however you like (without cutting it), and then gluing the ends together. It is the same as a knot you might tie with real string, except the ends are stuck together creating a closed loop. A circle, the "unknot," is a trivial knot. In the same way that a prime number is not divisible by any number other than 1 and itself, a prime knot can only be composed of itself and the unknot. I associate different knots with variations of the traditional Chinese tune, "Jasmine Flower." Variations, fresh takes on something old, create interest and beauty-- in math, music, and life.

The cover art is a chart of all prime knots with seven or fewer crossings. Technically, the unknot (circle) is not considered prime, in the same way the number "1" is neither prime nor composite. Knots can be classified by the number of crossings they have. The numbers below each knot signify the number of crossings, and the subscripts indicate different prime knots with each number of crossings. As indicated, there is only one prime knot with four crossings (4-1). There are two with five crossings (5-1, 5-2), etc.

The back cover depicts Reidemeister moves, "allowed" operations which do not "change" or increase the complexity of the knot. These moves include twisting a loop, sliding a loop over another loop, or sliding a loop over a crossing.

## KNOTS and JASMINE FLOWER

Each reharmonization in the Jasmine Flower Suite treats the traditional Chinese melody differently, using various keys, time signatures, instruments, and feels to capture certain moods. Roughly speaking, I wanted each arrangement to "sound" like the "shape" of a prime knot or related structure, with the harmonic density and ‘feel' of each piece mimicking the path taken to tie each knot. I associate different versions of "Jasmine Flower' with prime knots like trefoils, and their related structures including tori, tangles, braids, and wild knots.

As you listen to the pieces, try to envision the "shape" of each knot and think about the harmonies and  dissonances. Some feel round and beautiful, and others remind you of a tangled ball of Christmas lights.

Jasmine Flower (MoLiHua) in the original a capella vocal version is bare, pure, hollow yet full: the "unknot."

Jasmine Flower: Wild Knot
In knot theory, the more poorly-behaved knots are called "wild knots." They exhibit "pathological" behavior, are counterintuitive and atypical. In this arrangement, the three-horn introduction and the alto saxophone counterline to the melody employ very non-traditional counterpoint. I ignored traditional rules of voice leading and crossing. The overall density of the intro is very "thick." It is hard to trace the individual lines of the instruments, similar to the infinitely knotted loop structure of a wild knot.

Jasmine Flower: Trefoil
A trefoil is the simplest prime knot and the only knot with three crossings. Trefoils are very open and round. This particular arrangement contains a lilting rhythmic pattern in the piano which eventually gives way to a bass groove. Every two measures are divided into three attacks. The harmonic environment and chord changes, coupled with the repeated rhythmic groove create a very open feeling and a round quality. The placement of the tenor saxophone (melody) and alto saxophone (counterline) in the same sonic register creates a feeling of balance and equality between the two parts.

Jasmine Flower: Torus
A torus knot lies on the surface of a torus, which is a topological figure resembling a doughnut or life preserver. Although the knot lies on a surface of a round torus, tori knots themselves can become complex and very hard to visualize. The sparse arrangement in this piece features a repeating bass arpeggio containing roughly the same rhythm and note voicings for each chord. Torus knots can become rather dense and convoluted, which is why I chose to use an ominous minor key for this piece.

Jasmine Flower: Tangle
Tangle theory is the same as knot theory except that instead of closed loops, the two ends of the string are nailed down. This arrangement sounds "tangled," like fishing lines or a ball of Christmas tree lights. The melody is stated in the right hand of the piano, but the chords underneath are dissonant and unresolved. The melody is preserved and played in the right hand without a counterline, reminiscent of the closed loops with their ends tied town. The left hand chords and thin-yet-opaque texture symbolize the crossings and convoluted ways to "tangle" up the melody. Although seeking a resolution, the piece closes, never untangled.

Jasmine Flower: Braid
Braids in math are like taking strands of string, "braiding" them together like with hair, and then connecting the ends of to make closed loops. They are like circles which are lying next to each other but crossed and intertwined along the way. This arrangement is a "blues" version of Jasmine Flower. I employed a piano trio, a swing feel, and used a chord progression that starts out as a normal blues, but then veers off dramatically into its own world. At the end of the form, the regular blues progression resurfaces, though it ends on the IV chord instead of the I. The different musical lines are woven together in a beautiful braid.

Bass Prelude was inspired by one of my favorite piano pieces: Rachmaninoff's Prelude in G-sharp minor, Op. 32 No. 12. The lush melody in the left hand and ostinato in the right hand inspired the texture and mood of my piece. Here, the melody features the acoustic bass. The classical opening evolves into a jazz waltz, with various "classical" figures that recur.

Granada has gone through several different versions over the past few years, and is a place I would like to visit soon.

I called this tune Prime Knot because it reminded me of how beautiful these structures are.

Uniform Convergence, a more angular composition, certainly doesn't sound uniform or convergent. It pokes fun at this mathematical property of functions, a topic which took me a long time to understand in school.

Piano Prelude #1 originally existed as part of "The Patchwork Life," but as I wrote more, it became its own piece. This is my first work for solo piano, and bringing back my classical piano chops and recording this tune was very personally rewarding. I incorporated influences from some of my favorite Debussy pieces including "L'isle Joyeause" and "La Fille aux Cheveux de Lin."

The various sections of The Patchwork Life represent the patches of my own life. The piece is a celebration of the ups, downs, unexpected twists, and myriad of influences that make up my music, experiences, and life.

I wrote Perugia while sitting by a water fountain in Perugia, Italy at the end of the Umbria Jazz Festival 2009.

Eye Contact contains the album's main message, which is simple and human: "I ask you to get to know me, I hope you think I'm worth the time." It is a new arrangement of one of the first vocal songs I ever wrote, and is not about anyone in particular.