Knotoids



Updated 12th July, 2015

Here we present a list of distinct knotoids, introduced by Vladimir Turaev in [1], with up to 5 crossings. The list was produced a by computer search based on labelled immersion codes, a description of which is available here.

To represent a knotoid K using a labelled immersion code we add a shortcut that passes everywhere under K, forming K_ in Turaev's notation. Then, K_ is a knot for which we can write the labelled immersion code determined by numbering the semi-arc containing the leg of K as zero and proceeding in the direction from the leg to the head. We then identify the first crossing introduced by the shortcut by writing a ^ symbol after the crossing number in the immersion code. There is a unique semi-arc that enters this crossing as an under-arc with the orientation of K_ described above. Thus the ^ character uniquely identifies the semi-arc containing the head of K.

The programme that produced these tables is available as a source distribution here: vlist-release-12-7-15.tar

The following knotoids are distingished by Turaev's extended bracket polynomial, described in [1]. The diagrams below have been produced automatically using the draw programme. It has not yet been possible to create aesthetic normalized knotoid diagrams, where the leg is in the unbounded component of the diagram's compliment, using this approach.

k2.1 labelled immersion code: (-0 -2 -1^) / + + +
Extended bracket polynomial: −A10u2+A6u2+A4

k3.1 labelled immersion code: (-0 -2^)(1 3) / + + + +
Extended bracket polynomial: A−8−A−4−A−2u2+A2u2+1

k4.1 labelled immersion code: (-0 -2^)(-1 -4 -3) / + + + + +
Extended bracket polynomial: −A18u2−A16+A14u2+2A12−A8+A4

k4.2 labelled immersion code: (-0 -3 -1 -4 -2^) / + + + + +
Extended bracket polynomial: −A22u2+A18u2−A14u2+A10u2+A8

k4.3 labelled immersion code: (-0 -3 -2^)(-1 -4) / + + + + +
Extended bracket polynomial: −A18u2+A14u2+A12−A10u2−A8+A6u2+A4

k4.4 labelled immersion code: (-0 -2 -1)(-3 -5^ -4) / + + + - - -
Extended bracket polynomial: −A−6u−2−A6u2−A−4−A4+A−2u−2+A2u2+3

k4.5 labelled immersion code: $(-0 -2 -1)(3 5^ 4) / + + + - - -
Extended bracket polynomial: A20u4−2A16u4−2A14u2+A12u4+2A10u2+A8

k4.6 labelled immersion code: (-0 -3 -2)(1 5^ 4) / + + + + + -
Extended bracket polynomial: −A6u2−A−4u4+A2u2+u4+1

k4.7 labelled immersion code: (-0 -3 -2)(1 5^ 4) / + + + - + -
Extended bracket polynomial: A−14u2−2A−10u2−A−8u4+A−6u2+A−4u4+A−4

k4.8 labelled immersion code: (-0 -3 -2)(1 5^ 4) / + - + + - -
Extended bracket polynomial: −A16u4−A14u2+A12u4+A10u2+A8

k4.9 labelled immersion code: (-0 -3 -2)(1 5^ 4) / + - + - - -
Extended bracket polynomial: −A12u4−A10u2+A8u4+2A6u2+A4−A2u2

k5.1 labelled immersion code: (-0 -2 -1^)(-3 -5 -4) / + + + + + +
Extended bracket polynomial: A26u2−2A22u2−A20+A18u2+A16−A14u2+A10u2+A8

k5.2 labelled immersion code: (-0 -2 -1^)(-3 -5 -4) / + + + - - -
Extended bracket polynomial: −A−12−A−10u2+A−8+2A−6u2−A6u2−A−2u2+A2u2+1

k5.3 labelled immersion code: (-0 -2^)(1 4)(-3 -5) / + + + + + +
Extended bracket polynomial:  A10u2+A−8+A8−A6u2−A−4−2A4+2

k5.4 labelled immersion code: (-0 -2 -5 -3^)(1 4) / + + + + + +
Extended bracket polynomial: A14u2−2A10u2+2A6u2+A−4+A4+A−2u2−2A2u2−1

k5.5 labelled immersion code: (-0 2 5 -3 -1 4^) / + + + + + +
Extended bracket polynomial: −A12+2A8−A−6u−2+A6u−2−2A4+2A−2u−2−2A2u−2+2

k5.6 labelled immersion code: (-0 2 5 -3 -1 4^) / + + - + + -
Extended bracket polynomial: −A24−A22u−2+A20+2A18u−2−2A14u−2+A10u−2+A8

k5.7 labelled immersion code: (-0 2 -4 -1 5 3^) / + + + + + +
Extended bracket polynomial:  −A12−A10u−2+2A8−A−6u−2+2A6u−2−A4+2A−2u−2−2A2u−2+1

k5.8 labelled immersion code: (-0 -3^)(1 4 2 5) / + + + + + +
Extended bracket polynomial: A−20−2A−16−A−14u2+2A−12+A−10u2−A−8−A−6u2+A−4+A−2u2

k5.9 labelled immersion code: (-0 -3 -1 -4^)(2 5) / + + + + + +
Extended bracket polynomial: A14u2−2A10u2−A8+2A6u2+A−4+2A4−A2u2−1

k5.10 labelled immersion code: (-0 -3^)(1 5)(2 4) / + + + + + +
Extended bracket polynomial: A−16−A−12−A−10u2+A−8+A−6u2−A−4−A−2u2+A2u2+1

k5.11 labelled immersion code: (-0 -2 -1)(-3 -5)(4 6^) / + + + - - - -
Extended bracket polynomial: −A12u4−A10u2+2A8u4+3A6u2+A−4−A4u4+A4+A−2u2−3A2u2−1

k5.12 labelled immersion code: (-0 -2 -1)(3 5)(-4 -6^) / + + + - - - -
Extended bracket polynomial: −A18u2+2A14u2+2A12−2A10u2−3A8−A6u−2+A6u2+2A4+A2u−2

k5.13 labelled immersion code: (-0 -2 -1)(-3 6^ -5 4) / + + + + + + -
Extended bracket polynomial: −A18u2+2A14u2+A12u4+A12−2A10u2−2A8u4−A8+A4u4+A4+A2u2

k5.14 labelled immersion code: (-0 -2 -1)(-3 6^ -5 4) / + + + - + - -
Extended bracket polynomial: −A−6u2−A6u2−A−4u4−A4u4+A−2u2+A2u2+2u4+1

k5.15 labelled immersion code: (-0 -2 -1)(3 -6^ 5 -4) / + + + + + + -
Extended bracket polynomial: −A12−A10u2+2A8+A6u−2+2A6u2+A−4+A−2u2−A2u−2−2A2u2−1

k5.16 labelled immersion code: (-0 -2 -1)(3 - 6^ 5 -4) / + + + - + - -
Extended bracket polynomial: A20−2A16−A14u−2−A14u2+A12+A10u−2+A10u2+A8

k5.17 labelled immersion code: (-0 -2)(1 -3 -5 -4 6^) / + + + + + + -
Extended bracket polynomial: −A16+A12+A4

k5.18 labelled immersion code: (-0 -2)(1 5)(3 6^ 4) / + + + - + + -
Extended bracket polynomial: A−8+A−6u2−A−4+A4u4−2A−2u2+A2u2−u4+1

k5.19 labelled immersion code: (-0 -2)(1 5)(3 6^ 4) / + + + - - + -
Extended bracket polynomial: −A10u2+A8u4+2A6u2+A−4−A4u4+A4+A−2u2−2A2u2−1

k5.20 labelled immersion code: (-0 -2 -4)(1 5)(3 6^) / - - + - + - -
Extended bracket polynomial: −A18u2+2A14u2+A12−2A10u2−A8u4−A8+A6u2+A4u4+A4

k5.21 labelled immersion code: (-0 -2 -4)(1 5)(3 6^) / - - - - + - -
Extended bracket polynomial: A8−A−6u2−A4u4−A4+2A−2u2−A2u2+u4+1

k5.22 labelled immersion code: (-0 -3)(-1 6^ -5 2 4) / + + + + + + -
Extended bracket polynomial: A10u2+A8−A−6u2−A6u2−A4u4−2A4+2A−2u2−A2u2+u4+2

k5.23 labelled immersion code: (-0 -3)(-1 6^ -5 2 4) / + - + + + - -
Extended bracket polynomial: A−16+A−14u2−A−12−3A−10u2−A−8u4+2A−6u2+A−4u4+A−4

k5.24 labelled immersion code: (-0 -3)(-1 -6^ -5)(2 4) / + + + + + + -
Extended bracket polynomial: −A12−A10u−2+A8+2A6u−2+A6u2+A−4+A4−A2u−2−A2u2−1

k5.25 labelled immersion code: (-0 -3)(-1 -6^ -5)(2 4) / + + - + - + -
Extended bracket polynomial: −A24+2A20+A18u−2−2A16−2A14u−2−A14u2+A12+A10u−2+A10u2+A8

k5.26 labelled immersion code: (-0 -3)(-1 -6^ -5)(2 4) / + - + + + - -
Extended bracket polynomial: A−14u−2+A−12−2A−10u−2−2A−8+A−6u−2−A−6u2+2A−4+A−2u2

k5.27 labelled immersion code: (-0 -3)(-1 -6^ -5)(2 4) / + - - + - - -
Extended bracket polynomial: A8−A−6u−2−A6u2−A−4−2A4+2A−2u−2−A2u−2+A2u2+3

k5.28 labelled immersion code: (-0 4 -2 6^ -5 -3 1) / + + + + + + -
Extended bracket polynomial: A16−A12−A10u−2−A10u2+A6u−2+A6u2+A4

k5.29 labelled immersion code: (-0 4 -2 6^ -5 -3 1) / + + + + - + -
Extended bracket polynomial: −A18u−2−A18u2−A16+A14u−2+A14u2+3A12−2A8+A4

k5.30 labelled immersion code: (-0 4 -2 6^ -5 -3 1) / + + - + + - -
Extended bracket polynomial: A−10u−2+A−8−A−6u−2−2A−4−A−2u2+A2u2+2

The search algorithm used by the programme vlist sequentially considers all immersions with n crossings and assigns labels to those immersions in all possible combinations. The resultant candidates are then checked to see if they are equivalent to an immersion with fewer crossings (by a Reidemeister I or II move). Each distinct immersion gives rise to a number of candidate knotoid diagrams by reversing the above procedure for representing knotoids with labelled immersion codes. We consider the leg to be located on the semi-arc numbered zero by the immersion code and sequentially assuming each crossing to be the first crossing introduced by a shortcut between the head and the leg that passes everywhere under the knotoid. Each candidate is checked to see if it is a valid knotoid diagram and if so, the extended bracket polynomial is calculated.

One consequence of this approach is that having distinguished a set of knotoids with k crosings from immersions with n crossings, one cannot discount the possibility of finding another distinct knotoid with k crossings from an immersion having m > n crossings.

References:

[1] V. Turaev, Knotoids, arXiv:1002.4133v4

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