The Entangled Hearts Knot

Names and notations

Rolfsen notation

7.4

Dowker-Thistlethwaite notation

8 12 10 2 14 4 6

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Starting at the crossing with a "1" by it, we treat the "1" as the undercrossing and continue down to the next crossing with a "2" by it. Continuing this process, we label each crossing that we come to with the next integer until we come back to the beginning of the projection, at which point we have labelled each crossing with two numbers- one number for the undercrossing and one for the overcrossing. For our projection, we match up the odd numbers with the even numbers that share their crossings:

1 3 5 7 9 11 13
8 12 10 2 14 4 6

(Since each crossing for any projection of a knot will always have one even number and one odd number, we can list only the even numbers as they correspond to the odd numbers in ascending order.)
Also notice that for our projection, if we carry out this process, the even numbers always occur at the overcrossings. Thus, all the even numbers in our Dowker notation are positive by the convention of distinguishing this fact.

Conway notation

313

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This projection of the Entangled Hearts Knot is formed by starting with a 3 tangle, reflecting it across the NW and SE diagonal, and attaching it to a 1 tangle. Then, we reflect the result of this attachment (the 31 tangle) across the NW and SE diagonal and attach it to another 3 tangle.

Signed planar graph

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Each vertex corresponds to a point in every other region of the projection that would be shaded if the outside region were not shaded, and each edge corresponds to a crossing between the regions with vertices (in this projection, there are seven crossings). If we view each pair of vertex regions (between which a crossing occurs) one on top of the other, the overcrossing always occurs from southwest to northeast, so each edge of the signed graph has a + labeling.

Braid notation

σ_-1σ_-2σ_-1σ_-3σ_-2σ_-1σ_-1σ₂σ_-1σ₃σ₂σ₂σ₃σ_-1σ_-2

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We can rearrange the strands of our knot so that we obtain a projection as above. We have four strings for our closed braid representation, and we number these strings in ascending order starting from the left. Then we travel along the projection downward and label each crossing according to which strings cross and which one goes over: we write σ and in the subscript the number of the string which crosses from the left, keeping this number positive if the string from the left crosses over the string to its right and making it negative if the string from the left crosses under the string to its right. To determine the number to denote in the subscript for each crossing, we always look at the current position of the string in the braid. If we follow this procedure, we obtain the braid notation above.

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Knot properties

Alternating knot

Yes

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The crossings of the projection alternate between over and under, so the knot is alternating.

Tricolorable

Yes

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This projection is colored in such a way that at each crossing, either all three strands have the same color or they all have different colors - with the condition that at least two colors be used overall.

Amphicheiral

No

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According to Colin C. Adams' The Knot Book, the HOMFLY polynomial of the mirror image of a knot is obtained by replacing each l of the HOMFLY polynomial of the original knot with an l^-1. Doing so for the HOMFLY polynomial of the Entangled Hearts Knot, which can be seen further down on this page, we obtain m²(l^-1)^-2 + 2m²(l^-1) ^-4+ m²(l^-1) ^-6 +2(l^-1)^-4 - (l^-1) ^-8 = m²l² + 2m²l⁴ + m²l⁶ + 2l⁴ - l⁸, which does not equal m²l^-2 + 2m²l ^-4 + m²l ^-6 + 2l ^-4 - l ^-8, the HOMFLY polynomial of the Entangled Hearts Knot. Since the HOMFLY polynomial of the Entangled Hearts Knot and that of its mirror image are different, the Entangled Hearts Knot is not amphicheiral.

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Numerical knot invariants

Unknotting number

2

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If we change the two circled crossings of this standard projection of the Entangled Hearts Knot, we can obtain a projection of the unknot. This means that the knot's unknotting number is less than or equal to 2. The unknotting number of our knot cannot be 0 because this would imply that our knot is the unknot, which we have already discussed is not the case. It takes quite a bit more work to show that there is no projection of our knot such that we can change only one crossing to obtain a projection of the unknot. However, by other methods not discussed on this pet knot webpage, mathematicians have discovered that 2 is in fact the least number of crossing changes that must be performed, considering all the projections of our knot, to obtain a projection of the unknot.

Bridge number

2

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We turn our standard projection of the Entangled Hearts Knot approximately 90 degrees clockwise. Then we mark the orange points shown in the picture and imagine slicing a plane through the four orange points to cut our knot into two portions: an above-the-plane part and a below-the-plane part. Thinking of the strands as glued to the points but with the freedom to move around in space above or below the plane, we disentangle the strands of each part. Because we are able to disentangle the above-the-plane and below-the-plane parts, respectively, we know that we can obtain a projection of our knot in which the all the strands above the plane are respectively untangled and all the strands below the plane are respectively untangled. Thus, looking down onto our projection from an aerial view (treating the plane as the ground), we can see a projection with the (respectively untangled) strands of the above-the-plane part crossing over the (respectively untangled) strands of the below-the-plane part. Hence, because we have two strands above the plane, this aerial-view projection is one with two maximal overpasses. Since the bridge number of a knot is defined as the minimum number of maximal overpasses of all the projections of the knot, we then know that the bridge number of the Entangled Hearts Knot is less than or equal to 2.

We also know that knots with bridge number 1 are just unknots, and because we know that our knot is not the unknot (because of our knot's tricolorability property), the bridge number of our knot is not 1. Hence, the bridge number of our knot is 2.

Crossing number

7

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As Colin C. Adams writes in The Knot Book, Louis Kauffman, Kunio Murasugi, and Morwen Thistlethwaite proved independently in 1986 that an alternating knot in a reduced alternating projection of n crossings has crossing number n. The standard projection of the Entangled Hearts Knot with which we introduced our knot is a reduced alternating projection, and it has seven crossings. Thus, the crossing number of our knot is 7.

Genus

1

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To compute the genus of our knot, we construct a Seifert surface of it using Seifert's algorithm. We orient our knot and then remove crossings as follows: each undercrossing strand going into a crossing joins the part of the overcrossing strand coming out of the crossing, and the part of the overcrossing strand approaching the crossing joins the undercrossing strand coming out of a crossing. By removing crossings in that way, we create a certain number of Seifert circles, which we now connect to one another at the sites of the original crossings with half-twisted bands (twisted to the left if the overcrossing strand of the original knot crossing traveled to the left across the crossing, according to our orientation, and twisted to the right in the contrary situation).

Following Seifert's algorithm, we have constructed an orientable surface with the knot as its boundary! (Our surface is painted blue and green- the two white regions in the middle do not make up part of the surface. It is orientable in that we can paint it two colors so that the colors do not come in contact except on the boundary of the surface.)

By adding two vertices across each half-twisted band and connecting each of these pairs of vertices with an edge, we obtain a division the surface into six faces (one for each earlier Seifert circle), fourteen vertices (two at each crossing), and 21 edges (one at each crossing and two connecting each crossing to the next crossing, traveling by our orientation). Thus, the Euler characteristic of our surface is V-E+F = 14 - 21 + 6 = -1. As we mentioned, our surface has one boundary (our knot), so if we fill in the boundary with a disk, we obtain a surface with Euler characteristic equal to 0. A surface with no boundary with Euler characteristic of 0 is a torus, which has genus 1 (a genus corresponds to the number of holes in a surface).

A genus of a knot is defined as the least genus of any Seifert surface of that knot, so we know that the genus of our knot is less than or equal to 1. However, our knot does not have genus 0 because genus-0 knots are unknots (a surface with genus 0, a sphere, is capped off by a disk, which has a boundary of an unknot) and we discovered earlier that by the tricolorability property of our knot, it is not the unknot. Thus, the genus of the Entangled Hearts Knot is 1.

Span of the bracket polynomial

28

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In The Knot Book/, Adams states and provides the proof of the lemma that if a knot has a reduced alternating projection of //n crossings, then the span of a knot, or difference between the highest and lower powers of the bracket polynomial of the knot, is 4n. The standard projection of the Entangled Hearts Knot is a reduced alternating projection and has 7 crossings. Thus, the span of the Entangled Hearts Knot is 4(7) = 28.

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Knot polynomials

Jones polynomial

-q⁸ + q⁷ - 2q⁶ + 3q⁵ - 2q⁴ + 3q³ - 2q² + q

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Check out Knot Atlas website: http://katlas.org/wiki/7_4.

Alexander polynomial

4t - 7 + 4t ^-1

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See Knot Atlas website at http://katlas.org/wiki/7_4.

HOMFLY polynomial

z²a ^-2 + 2z²a ^-4 + z²a ^-6 + 2a ^-4 - a ^-8

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See Knot Atlas website: http://katlas.org/wiki/7_4.