Table of contents

The LooPindex is a digital catalog containing the pinning data of all irreducible, indecomposable multiloops in the unoriented sphere with at most 12 complementary regions. There are a total of 1097 such.

Multiloops by regions and strands

The multiloops are organized first by number of complementary regions, and then by number of strands. For instance, the multiloop $8^3_2$ has 8 regions, 3 strands, and is the second on our list of such multiloops. By region, the number of multiloops total follows the sequence [oeis.org, A113201]. The number of loops (multiloops having one strand) follows the sequence [oeis.org, A264759]. The sequences corresponding to a fixed higher numbers of strands have not yet been found on oeis.org.

• 4 regions (1 multiloop)
• 2 strands (1 multiloop)
• 5 regions (1 multiloop)
• 1 strand (1 loop)
• 6 regions (2 multiloops)
• 1 strand (1 loop)
• 2 strands (1 multiloop)
• 7 regions (3 multiloops)
• 1 strand (2 loops)
• 2 strands (1 multiloop)
• 8 regions (9 multiloops)
• 1 strand (3 loops)
• 2 strands (4 multiloops)
• 3 strands (2 multiloops)
• 9 regions (18 multiloops)
• 1 strand (10 loops)
• 2 strands (7 multiloops)
• 3 strands (1 multiloop)
• 10 regions (62 multiloops)
• 1 strand (27 loops)
• 2 strands (27 multiloops)
• 3 strands (7 multiloops)
• 4 strands (1 multiloop)
• 11 regions (198 multiloops)
• 1 strand (101 loops)
• 2 strands (77 multiloops)
• 3 strands (19 multiloops)
• 4 strands (1 multiloop)
• 12 regions (803 multiloops)
• 1 strand (364 loops)
• 2 strands (341 multiloops)
• 3 strands (85 multiloops)
• 4 strands (12 multiloops)
• 5 strands (1 multiloop)

Other interesting classes of multiloops

• The main page shows all multiloops with 12 or fewer regions for which the minimal region-degree is strictly greater than 2 (necessarily 3). There are only 6 of them. By number of regions, the number of such multiloops follows the sequence [oeis.org, A078666].
• The class of multisimple multiloops is interesting. There are 65 of them with 12 or fewer regions. By number of regions, the number of such multiloops follows a sequence 1, 0, 1, 0, 4, 0, 10, 0, 49, 0, 336, 0, 3154... which has not yet been found on oeis.org.