An Algebra of Knots

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 T₁(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 (T₁ⁿ) of n turns forward along the positive direction (O towards F) of the knot, (b) round the loop starting at T₁, and finally (c) between T₁ ⁿ and the end.  The ‘turns’ in question come about when the free end of the rope is wrapped round an earlier (‘passive’) section of the rope. The number of turns can be positive (clockwise) or negative (anti-clockwise) but in each case the turns are to be created in a forward-increasing-spiral round the given, passive rope. This is essential, because a backward-increasing spiral would not bring the free end of the rope through the loop:  in which case the ‘wrap’ would not be locked into place. (This is topology and the same number of turns of the looped section in reverse would result in the ‘wrap’ disappearing.) 

AMALGAMATING WRAPS 
A possibility of simplifying the algebraic representation of a knot occurs when two adjacent wraps T_a and T_a+1 are situated on a common length of string and when no subsequent wrap has been interposed on this length of string between them. A simple example of this is shown in Diagram 1. Here T₂ is located on the first loop x₂ of a knot and its three positive turns can be, in effect, amalgamated with the two positive turns of T₁ producing a new T₁ with five positive turns as shown in the RH side of Diagram 1.

[Diagram 1]    

More generally if  T_pⁿ is a wrap anywhere on a knot and is followed by T_p+1^-m on a common section of rope which is not subsequently partitioned in the construction of the complete knot, the effect will be merely to reverse m of the positive turns introduced by T_p ⁿ. So in the notation given the pth wrap becomes T_p^n-m and the suffixes of all subsequent Ts (wraps) reduce by 1.  The suffixes of all the subsequent xs reduce by 2.

A generalised statement of this principle is needed which will apply to the pth 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_r+1 followed by x_r+2. (See Diagram 2.) 

[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_pⁿ (x_r+1 T_p+1(x_r+2^–m) x_r+3) x_r+4 …

After amalgamation the state of affairs is represented by

… x_r T_p(x_r+1) ^n-m x_r+2 ….

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

… x_r T_pⁿ (x_r+1 T_p+1(x_r+2^–m) x_r+3) x_r+4 …  =   … x_r T_p(x_r+1) ^n-m x_r+2 ….

Incidentally r = 2p -1  because each additional wrap increases the x suffix by 2 and x₁ 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:

1. 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.
2. 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 + 2n added to them.
3. 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 K², K³, K⁴, … 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 –R turns. This procedure is then repeated N-1 times on the resulting (reduced) knot successively… until a plain unwrapped rope results.

CHRISTOPHER ORMELL 1^st October 2022