Six Lines

It is by no means so easy to show that there are in all 19 types of interlacings of six lines (this theorem was proved by Mazurovski [4], [5] in 1987). It is no longer possible to distinguish between nonisotopic interlacings using only the linking numbers of the triples of lines in each interlacing. To prove that the isotopy classes are really distinct one has to perform computer calculations of more complicated invariants of the interlacings. Before describing Mazurovski's basic results in more detail, we give some definitions.

We shall need a construction proposed by Mazurovski to characterize interlacings of lines. Given a permutation $\sigma$, he constructs a corresponding interlacing defined up to isotopy. Let $l$ and $m$ be oriented skew lines whose linking number is $-1$.

We mark off $k$ points on each line $l$ and $m$, and denote them by $A_1,\dotsc,A_k$ and $B_1,\dotsc,B_k$, where moving form point to point with increasing indices takes us in the direction of the line's orientation. Now, given a permutation $\sigma$ of $\{1,\dotsc,k\}$, we construct an interlacing of $k$ lines by joining $A_i$ to $B_{\sigma (i)}$. Following Mazurovski, we shall denote this interlacing of the $k$ lines $A_1B_{\sigma (1)},\dotsc,A_kB_{\sigma (k)}$ by the symbol $jc(\sigma )$. Interlacings which are isotopic to an interlacing constructed in this way are said to be isotopy join.

Exercise Which of the interlacings encountered above are isotopy joins? Show that all interlacings of five or fewer lines are isotopy joins.

Mazurovski [6] showed that, if we want to prove that two interlacings of six lines are not isotopic or if we want to determine the isotopy class of a given interlacing of six lines, it is sufficient to use the polynomial invariant of framed links in $\mathbb{R}P^3$ which was introduced by Drobotukhina [7]. This invariant generalizes the Kauffman polynomial of links in $\mathbb{R}^3$.

Return to the classification of interlacings of six lines. Of the 19 types, 15 consist of isotopy join interlacings. The remaining four are the interlacing types $M$ and $L$ in Figures 26 and 27,

Figure 26: Interlacing $M$

Figure 27: Interlacing $L$

and their mirror images $M'$ and $L'$. Here $M$ and its mirror image $M'$ cannot be distinguished by means of the linking numbers of the triples in the interlacings. But they can be distinguished using Drobotukhina's polynomial invariant, which for $M$ is equal to

$$-A^{15}+6 A^{11}+6A^9-5A^7-6A^5+10A^3+16A$$

$$+ A^{-1}-10A^{-3}+10A^{-7}+5A^{-9},$$

and for $M'$ is equal to

$$5A^9+10 A^7-10A^3+A+16A^{-1}+10A^{-3}$$

$$-6 A^{-5}-5A^{-7}+6A^{-9}+6A^{-11}-A^{-15}.$$

Similarly, $L$ cannot be distinguished from the interlacing $jc(1,2,5,6,3,4)$ by means of the linking numbers, but these two interlacings do have different polynomial invariants: for $L$ it is

$$A^{17}-5A^{13} +15A^{9}+10A^{7}-13A^{5}-12A^{3}+15A$$

$$+22A^{-1}-A^{-3}-12A^{-5}+A^{-7}+8A^{-9}+3A^{-11}$$

and for $jc(1,2,5,6,3,4)$ it is

$$A^{13}+A^{11}+4A^{7}+7A^{5}+3A^{3}+2A^{-1}+5A^{-3}+3A^{-5} +2A^{-9}+3A^{-11}+A^{-13}.$$

The derived interlacing of $L$ coincides with $L$ itself. The same holds for the mirror image $L'$ of $L$, the interlacings $M$ and $M'$, and also the amphicheiral interlacing $jc(1,3,5,2,6,4)$. The interlacings $jc(1,2,4,6,3,5)$ and $jc(5,3,6,4,2,1)$ (which are mirror images of one another) both have the same derived interlacing, namely, an amphicheiral interlacing of five lines (which coincides with its own derived interlacing). The remaining types of interlacings of six lines are completely decomposable. Of those 12 types, two are the amphicheiral interlacings

$$\langle \langle +3\rangle ,\langle -3\rangle \rangle =jc(1,2,3,6,5,4)$$

and

$$\langle \langle -\langle 1\rangle , \langle +2\rangle \rangle ,\langle +\langle 1\rangle ,\langle -2\rangle \rangle \rangle =jc(1,2,4,6,5,3),$$

and the ten others can be divided into pairs of nonamphicheiral interlacings, each pair consisting of an interlacing and its mirror image:

$$\langle -6\rangle =jc(1,2,3,4,5,6),\qquad\langle +6\rangle =jc(6,5,4,3,2,1);$$

$$\langle \langle +2\rangle ,\langle -4\rangle \rangle =jc(1,2,3,4,6,5),$$ (1)

$$\langle \langle +4\rangle ,\langle -2\rangle \rangle =jc(5,6,4,3,2,1);$$ (2)

$$\langle +\langle -3\rangle ,\langle -2\rangle ,\langle 1\rangle \rangle =jc(1,2,3,5,6,4),$$ (3)

$$\langle -\langle +3\rangle ,\langle +2\rangle ,\langle 1\rangle \rangle =jc(4,6,5,3,2,1);$$ (4)

$$\langle -\langle +2\rangle ,\langle +2\rangle ,\langle -2\rangle \rangle =jc(1,2,4,3,6,5),$$ (5)

$$\langle +\langle +2\rangle ,\langle -2\rangle ,\langle -2\rangle \rangle =jc(5,6,3,4,2,1);$$ (6)

$$\langle +\langle -2\rangle ,\langle -2\rangle ,\langle -2\rangle \rangle =jc(1,2,5,6,3,4),$$ (7)

$$\langle -\langle +2\rangle ,\langle +2\rangle ,\langle +2\rangle \rangle =jc(4,3,6,5,2,1).$$ (8)