**The previous blog in this series (No. 28) introduced a construction protocol for knots which serves, in effect, as a way of classifying them mathematically. At first sight the construction diagrams arising from this protocol look much neater than typical knots. But the new protocol covers all possible sequences of finite wraps, and one has to imagine what they would look like when thoroughly dis-shevelled. They would look indeed like the messy tangles we call ‘extended knots’!**

**In this blog we look at the kind of abstract notation which naturally emerges as a way of representing such constructions. Some rules can be observed which may presage the arrival of a new algebraic domain.**

In Blog 28 we showed how the symbolic device: O x T_{1}(x) x F, where each ‘x’ stood for a length of open rope, could represent the simplest —single wrap— range of ‘knots’. The rope was divided into three sections (a) from the start to the first wrap (T_{1}* ^{n}*) of

**AMALGAMATING WRAPS**** **

A possibility of simplifying the algebraic representation of a knot occurs when two adjacent wraps T* _{a}* and T

[Diagram 1]

More generally if T* _{p}^{n}* is a wrap anywhere on a knot and is followed by T

A generalised statement of this principle is needed which will apply to the *p*th wrap and the *p*+1^{th} wrap on any common section of rope which has not subsequently been partitioned by a later wrap. Let the common section of rope be represented by x* _{r}* followed by x

[Diagram 2]

In general after amalgamation of the two wraps, the suffixes on the xs change in the following manner. The initial state of affairs shown in Diagram 2 is represented by

… x* _{r}* T

After amalgamation the state of affairs is represented by

… x* _{r}* T

Hence we have a simplification of a section of the algebraic representation as follows:

… x* _{r}* T

Incidentally *r* = 2*p* -1 because each additional wrap increases the x suffix by 2 and x_{1} is the initial section of rope starting at O.

**ADDING KNOTS**** **

Given an initial knot K and a new knot G it is possible to “add” G to K producing a composite knot (K+G) which is the knot G replicated and added on to the end (F) of knot K. See Diagram 3.

[DIAGRAM 3]

The algebraic expression for knot K+G is a conjunction of the two expressions (with a single x) with these changes:

- The wrap numbers of the latter part of the composite knot (i.e. the old knot G) have
*n*added to them, where*n*is the number of wraps in knot K. - The wrap superscripts of the latter part of the composite knot (i.e. the old knot plus G) have
*n*added to them and the x suffixes of the latter part of the old knot + G have 1 + 2*n*added to them. - The wrap turn superscripts of the latter part of the composite knot (the old knot G) remain the same.

This form of ‘addition’ is self-evidently not commutative. K+G clearly is not equal to G+K, unless G = K..

**MULTIPLYING KNOTS **** **

The knot formed when each x of knot K is replaced by a copy of knot G —with the suffices changed according to the same principles as above— can be treated as K multiplied by G. This form of multiplication is also clearly not commutative. The algebraic expression for K times G is formed by replacing each x of knot K by a copy of the expression for knot G while increasing all the suffixes in the new version following the same principles as (a), (b), (c) above.

By this means it is possible to form expressions for K^{2}, K^{3}, K^{4}, … etc.

There is no knot which, when added to K will reduce it to an unknotted rope.

**SUBTRACTION AND DIVISION OF KNOTS **** **

This is only possible as a reverse version of what has been previously added or multiplied.

**A PROTOCOL FOR DISENTANGLING KNOTS **** **

This is achieved by appropriately processing each wrap in turn starting with the final wrap T* _{N}^{R}*. The free end of the rope F where it emerges from the last wrap is subjected to –

**CHRISTOPHER ORMELL 1 ^{st} October 2022**