Post making contest 6.0

:woahtermelon:

Passes no rules

:dog:

Passes rule 2

In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself (crossing switch) to untie it. If a knot has unknotting number {\displaystyle n} n, then there exists a diagram of the knot which can be changed to unknot by switching {\displaystyle n} n crossings.[1] The unknotting number of a knot is always less than half of its crossing number.[2]

Any composite knot has unknotting number at least two, and therefore every knot with unknotting number one is a prime knot. The following table show the unknotting numbers for the first few knots:


Trefoil knot
unknotting number 1



Figure-eight knot
unknotting number 1



Cinquefoil knot
unknotting number 2



Three-twist knot
unknotting number 1



Stevedore knot
unknotting number 1



6₂ knot
unknotting number 1



6₃ knot
unknotting number 1



7₁ knot
unknotting number 3

In general, it is relatively difficult to determine the unknotting number of a given knot. Known cases include:

The unknotting number of a nontrivial twist knot is always equal to one.
The unknotting number of a {\displaystyle (p,q)} (p,q)-torus knot is equal to {\displaystyle (p-1)(q-1)/2} (p-1)(q-1)/2.
The unknotting numbers of prime knots with nine or fewer crossings have all been determined.[3] (The unknotting number of the 1011 prime knot is unknown.)
Contents
Other numerical knot invariants Edit

Crossing number
Bridge number
Linking number
Stick number
See also Edit

Unknotting problem
References Edit

^ Adams, Colin Conrad (2004). The knot book: an elementary introduction to the mathematical theory of knots. Providence, Rhode Island: American Mathematical Society. p. 56. ISBN 0-8218-3678-1.
^ Taniyama, Kouki (2009), "Unknotting numbers of diagrams of a given nontrivial knot are unbounded", Journal of Knot Theory and its Ramifications, 18 (8): 1049–1063, MR 2554334, doi:10.1142/S0218216509007361.
^ Weisstein, Eric W. "Unknotting Number". MathWorld.
External links Edit

"Three_Dimensional_Invariants#Unknotting_Number", The Knot Atlas.
Last edited 7 days ago by Hyacinth
RELATED ARTICLES
Knot theory
study of mathematical knots
List of mathematical knots and links
Wikimedia list article
Crossing number (knot theory)
integer-valued knot invariant; least number of crossings in a knot diagram
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I'm gonna say this post passes rule 1. It was a little tricky to call for reasons but I think it passes. Also rule 2 as well.

Alpha Beta Cranberry Delta

You say Gamma, I say Cranberry.

I've finally taken the time to really dig in and understand SU(2) and SO(3), and I have to say, it's fucking awesome. This is definitely one of my new favorite things. I mean look at this shit, somehow numbers like this:

[a + bi, -c+di]
[c + di, a - bi]

Are the same thing as numbers like this (!):

w + xi + yj + kz

Which are the same thing as good ol' ordinary rotation matrices (but twice!):

[angles, numbers, sin()]
[theta, stuff, 0]
[cos(-x), -1, 3] (rotation matrices are a pain to write)

Honestly even just the fact that the first type stays as the first type is really cool, let alone that it's a valid interpretation of quaternions. From their I knew that quaternions could represent positions/rotations in 3D space, but I had no idea that there was that extra half, or that that extra sign was what led to spinors, and in turn, the entire concept of half-integer spin in general, and from there, the concepts of mass and the pauli exclusion principle!!!!!!!

and since the whole SU(2)/SO(3) thing only really happens in 2 & 3 dimensions, I think that means that half-integer spin, and therefore massive particles, might only be possible in universes with three dimensions! or vice versa, their very existence is what the dimensions come from in the first place


fuck me this is so cool

I've finally taken the time to really dig in and understand SU(2) and SO(3), and I have to say, it's fucking awesome. This is definitely one of my new favorite things. I mean look at this shit, somehow numbers like this:

[a + bi, -c+di]
[c + di, a - bi]

Are the same thing as numbers like this (!):

w + xi + yj + kz

Which are the same thing as good ol' ordinary rotation matrices (but twice!):

[angles, numbers, sin()]
[theta, stuff, 0]
[cos(-x), -1, 3] (rotation matrices are a pain to write)

Honestly even just the fact that the first type stays as the first type is really cool, let alone that it's a valid interpretation of quaternions. From their I knew that quaternions could represent positions/rotations in 3D space, but I had no idea that there was that extra half, or that that extra sign was what led to spinors, and in turn, the entire concept of half-integer spin in general, and from there, the concepts of mass and the pauli exclusion principle!!!!!!!

and since the whole SU(2)/SO(3) thing only really happens in 2 & 3 dimensions, I think that means that half-integer spin, and therefore massive particles, might only be possible in universes with three dimensions! or vice versa, their very existence is what the dimensions come from in the first place


fuck me this is so cool

Passes rule 2, this is the only rule that has been passed in the above posts.

pobody's nerfect!

nobody passed anything

Not even gas?

pobody's nerfect!

Passes no rules

nobody passed anything

Not even gas?

Passes rule 3

ABCDEFGHIJKLMNOPQRSTUVWXYΩ

ABCDEFGHIJKLMNOPQRSTUVWXYΩ

No rules are passed though I approve of the use of the (semi) canon alphabet.

I kick ur rules

Winner winner chicken dinner

I'm just gonna... quietly... change rule 2 just a touch for the reason that whoops i made it too restrictive.

Posts that now pass rule 2 that didn't:

• 1378
  
• 1425
  
• 1428
  
• 1429
  
• 1433
  

Posts that did pass rule 2 but now don't:

• 1398
  

I kick ur rules

oof ouch not my rules. you don't pass any here

Winner winner chicken dinner

no rules passed

:<dog>:

>:(

:<dog>:

No rules passed

>:(

Ditto

:dog:

:dog:

no rules passed

:frick:

five is bigger number than four.

well, most of the time

rainbow

:frick:

Passes no rules

five is bigger number than four.

well, most of the time

Passes no rules

Passes no rules

rainbow

Good post. Passes rule 3.

woah.



I'm floating.