The Love Knot

Names and notations

Rolfsen notation

7₃

Dowker-Thistlethwaite notation

8 10 12 14 4 2 6

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Following from 1, the Dowker-Thistlethwaite Notation can be easily derived from this projection.

Conway notation

43

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This projection shows the 4 half twists on the top, and then the 3 on the bottom.

Braid notation

б₁ б₁б_-2 б₁ б₁ б₂ б₂б₁ б₁

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Here is a picture of my knot drawn from closing the braid given by the braid notation.

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Knot properties

Alternating knot

Yep!

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This knot is definitely alternating because its crossings go between over and under all of the way through, regardless of orientation.

Tricolorable

Not Tricolorable

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Type justification here

Put a ">(space)" in front of each new line to create the box (even on empty lines and in front of images).

You can type mathematics inline, $f(x)=\sin x$, but not in equation mode.

Amphicheiral

Yes/No

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Type justification here

Put a ">(space)" in front of each new line to create the box (even on empty lines and in front of images).

You can type mathematics inline, $f(x)=\sin x$, but not in equation mode.

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Numerical knot invariants

Unknotting number

2

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Type justification here

Put a ">(space)" in front of each new line to create the box (even on empty lines and in front of images).

You can type mathematics inline, $f(x)=\sin x$, but not in equation mode.

Bridge number

2

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Type justification here

Put a ">(space)" in front of each new line to create the box (even on empty lines and in front of images).

You can type mathematics inline, $f(x)=\sin x$, but not in equation mode.

Crossing number

TBD

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Type justification here

Put a ">(space)" in front of each new line to create the box (even on empty lines and in front of images).

You can type mathematics inline, $f(x)=\sin x$, but not in equation mode.

Genus

TBD

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Type justification here

Put a ">(space)" in front of each new line to create the box (even on empty lines and in front of images).

You can type mathematics inline, $f(x)=\sin x$, but not in equation mode.

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Knot polynomials

Jones polynomial

−q9 + q8−2q7 + 3q6−2q5 + 2q4−q3 + q

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This website is a good reference for this information: 7_3 - Knot Atlas

Alexander polynomial

2t2−3t + 3−3t−1 + 2t−2

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This website is a good reference for this information: 7_3 - Knot Atlas

HOMFLY polynomial

z4a−4 + z4a−6 + 3z2a−4 + 3z2a−6−z2a−8 + a−4 + 2a−6−2a−8

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This website is a good reference for this information: 7_3 - Knot Atlas