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# connected sum

The *connected sum of knots* $K$ and $J$ is a knot, denoted by $K\#J$,
constructed by removing a short segment from each of $K$ and $J$ and joining each free end of $K$ to a different free end of $J$ to form a new knot. The connected sum of two knots always exists but is not necessarily unique.

The *connected sum of oriented knots* $K$ and $J$ is a connected sum of knots which has a consistent orientation inherited from that of $K$ and $J$. This sum always exists and is unique.

###### Example.

Suppose $K$ and $J$ are both the trefoil knot.

By one choice of segment deletion and reattachment, $K\#J$ is the quatrefoil knot.

Related:

KnotTheory

Synonym:

knot sum

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

57M25*no label found*

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## Comments

## Connected sum graphics

After mps's correction to make the knot graphics "prettier" (meaning smoother with no unnecessary cusps) I thought I could just redraw them by hand. Unfortunately, my hands just aren't as steady as they used to be. In exchange for no arthritis pain I pay the price in shakiness and unsteadiness. I hope a young man with drawing skills will pick up this entry and make much better looking graphics. I don't mind if you delete my graphics, though you may want to keep them for reference until you get the better graphics in place.

## Re: Connected sum graphics

You might look at

http://www.dpmms.cam.ac.uk/~al366/xytutorial.html

for examples of drawing knots with xypic.

Roger

## Re: Connected sum graphics

That all looks very nice. In the graphics sandbox I tried the second diagram from http://www.dpmms.cam.ac.uk/~al366/braidtutorial/Knots.html I added \usepackage[all, knot]{xy} to the preamble like it says but when I clicked preview I got an unexpected Noosphere error. (The preamble change probably did not stick then).

## Re: Connected sum graphics

Did you try just uncommenting the usepackage{xypic} in the standard preamble?

## Re: Connected sum graphics

It was already uncommented for something else, which then would mean usepackage[all, knot]{xy} is redundant. I tried again, same helpful unexpected Noosphere error.

## Lie superalgebras

Hello, I am a new user. I have been working in signal processing data transform design, currently for applications in geophysics. One of the transform output spaces that I have created is symmetric. The definition that I have (and understand!) for a Lie algebra is as follows:

A non-associative algebra is said to be a Lie algebra if it's multiplication obeys the Lie conditions

1) x squared equals zero and

2) (xy)z+(yz)x+(zx)y=0 (Jacobi Identity)

Given this definition, can anyone add the other conditions that a Lie superalgebra must satisfy, in the same kind of mathematical language? Your definition on the web-site will take me a long time to understand, as I have not read a great deal of abstract algebra.

Regards,

Halfamatician

## Re: Lie superalgebras

Check this out

http://planetmath.org/encyclopedia/LieSuperalgebra3.html