Tangles and computational algebraic geometry

What is a Tangle?

A Tangle is a $C^1$ space curve formed by the union of quarter-circles with the same radius.

Tangles and algebraic geometry

Theorem (Prowell, T.)

One-to-one correspondence between $n$-Tangles and sequences of vectors $\mathbf v_0,\ldots,\mathbf v_{n-1}\in\mathbb R^3$ satisfying

• $\|\mathbf v_i\| = r$
• $\mathbf v_0 + \cdots + \mathbf v_{n-1} = 0$
• $\mathbf v_i \cdot \mathbf v_{i+1} = 0$

System of homogeneous polynomial equations in $3n + 1$ variables $\implies$ "Tangle variety" in $\mathbb P^{3n}$

Simplification

Assume $r = 1$, $\mathbf v_0 = \mathbf i$, and $\mathbf v_1 = \mathbf j$.

Now in $\mathbb A^{3n -6}$.

Macaulay2

Software system for research in algebraic geometry and commutative algebra.

i3 : affineTangleVariety 4

o3 = ideal (v   , v    + 1, v   , v   , v   , v    + 1)
             3,2   3,1       3,0   2,2   2,1   2,0

o3 : Ideal of QQ[v   ..v   ]
                  2,0   3,2

$\implies \mathbf v_2 = -\mathbf i$   and   $\mathbf v_3 = -\mathbf j$     (circle!)

No 5-Tangles

i4 : affineTangleVariety 5

o4 = ideal 1

o4 : Ideal of QQ[v   ..v   ]
                  2,0   4,2

What about 6-Tangles?

i6 : betti affineTangleVariety 6

            0  1
o6 = total: 1 16
         0: 1  6
         1: .  7
         2: .  3

o6 : BettiTally

Minimal associated primes

i10 : dim \ minimalPrimes affineTangleVariety 6

o10 = {1, 0, 0, 0}

o10 : List

1 curve, 3 points

Complex-valued 6-Tangle

i12 : last minimalPrimes affineTangleVariety 6

                                                                                                             
o12 = ideal (v    + 2, v   , v    + 2v   , v    - 3, v    + 2, v    - 2v   , v    + 2, v    - 3, v    + v   ,
              5,1       5,0   4,2     5,2   4,1       4,0       3,2     5,2   3,1       3,0       2,2    5,2 
      ------------------------------------------------------------------------------------------------------------
                       2
      v   , v    + 2, v    + 3)
       2,1   2,0       5,2

o12 : Ideal of QQ[v   ..v   ]
                   2,0   5,2

Other two points

Reflections of one another

The curve

$\|\mathbf v_2\| = 1$   and   $\mathbf j\cdot\mathbf v_2 = 0$ $\implies$

$\mathbf v_2=\langle\cos\theta, 0,\sin\theta\rangle$

Visualizing the curve

$\theta = \frac{\pi}{2}, \frac{3\pi}{4}, \pi$

Thank you!