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1 I o uc io e ai c osu es a e use u o e ese i g k o s [1-3] oge [ ] usi g y e-II ei emeis e mo es as gi e a ew oo a eac k o ca e e e- se e y a ai c osu e O y a ew a es o e ai wo s o k o s a e ee u is e [ ] o es [ ] as gi e ai wo s o a k o s wi u o 1 c ossi gs o o es equi i g ai i e u o e c ossi g um e C, o c ossi g i e o a k o is e sma es um e o c ossi gs a occu i a y o ec io o e k o e ai i e o a k o o i k is e mi imum um e o s i gs i a ai ee e o e ese e k o o i k y e ai c osu e I e a u a io o o es [ ] e ai wo e essio s a e o ee o imi e ye W i e o me o as ee o ose so a [1 ] o i e mos eco omic ese a io o ai s o k o s e e a ea s a ossi i i y o s a a i e sys ema ica y wo s o e sim e k o s e ( C) o us k o s a e e c ossi g um e C o e e sim e k o s a e e wis k o s w ic a e a ec o emic egio co ai i g C - c oss- i gs i e wo e ike i e ( C - o us k o a a e ocke y a i e ock wi wo c ossi gs [5-7] We e e o e c assica k o esig a io a e k o iag ams o o se [5] e ious y [1 3] a ai ge e a o bn was e i e so a i yie e a ega i e c ossi g We e i e e ai ge e a o bn, so a i o uces a osi i e c ossi g [ -1 ] As i e [ ] we w i e e ai wo y a s o a o a io i a ai wo a ac o is w i e o e ai ge e a o a i+ o b+ wi a a u a p. e ai wo s w(Ci), wi i =1,2,3, o wis k o s ca e s a a i e y e use [ - 3 -1 ] o ocks A( a D(n), wi a u a 1
e Co way o a io [13] as i ica e i o se iag ams [5] wi o ow a e e ai wo a a comma o eac k o
( 79 680 M. S ff z ń
2. t n t th dd n b r f r n In the classical knot labeling [1-6], if the crossing number C is odd, the (2, C torus knot is labeled C1. The first type twist knot which has in a plectonemíc region C — 2 crossings interwoven like in the (2, C — 2) torus knot, and is locked by an interlock containing two crossings [5-7], is labeled C and has all its C crossings of the same sign [5-7]. The second type twist knot which is locked by an interlock of three crossings [5], is labeled C3 If C = 2n + 1, with natural n, the braid word for the (2, C torus knot with positive crossings is [2, 3]: and, in the case of negative crossings, is
For the twist knots C with low odd crossing numbers the braid words can be written [2] as
and, by induction on the number of crossings, for natural n> 1,
Since the length of each of the three blocks A(n), (A(n)) -1 , and (n de- pends linearly on the number of crossings, an increase in the crossing number C by two, i.e. an increase in the number n by one, enlarges the length of the braid word by three. The braids for the second type twist knots C3 with odd number C of cross- ings, including the three crossings of the interlock, have braid index three, and the braid words are [2, 3]:
and, for natural n> 2,
The braid words for knots 73 and 93 conform in crossing signs to the re- spective knot diagrams of Rolfsen [5, 13, 14]. Knots equivalent to braids of index three have been illustrated also by diagrams of Akutsu et al. [3]. Perko [15], in his diagrams of knots with crossing number 11, labeled the twist knots by labels which differ from the standard of Rolfsen notation [5, 13, 14, 16] used here. Braids for Twist Knots 681
. t n t th v n n b r f r n
If the crossing number C of knot is even, the first type twist knot which has in a plectonemic region C — 2 crossings interwoven like in a torus knot, and has an interlock containing two crossings of the other sign [5-7], is labeled C1 The second type twist knot, locked by an interlock with three crossings [5], is labeled C2, and the next type twist knot, locked by an interlock with four crossings [5], is labeled C3. The braid words for the twist knots C1, with low even crossing numbers, can be written [2] as
and, by induction on the number of crossings, for natural n> 2,
Since the length of each of the three blocks A(n - 1), (A(n - 1)) -1 , and D(n - 1), depends linearly on the number of crossings, an increase in the crossing number C by two, i.e. an increase in the number n by one, enlarges the length of the braid word by three. Knot 61 is called the stevedore knot [16]. 682 M S ff z ń
The braids for the second type twist knots C2, with even crossing numbers C see Fig. 1, have the braid index three, and the braid words are [2, 3]:
and, for natural n> 2,
The braid words for the third type twist knots C3, with low even crossing numbers C = 2n, with n> 3, can be written [2] as
and, by induction on the number of crossings, for natural n> 3,
The amphicheiral [5, 14, 18, 19] knot 63 is not a twist knot, it has Conway notation label [13] of a form distinct from that of the amphicheiral knot 83, and has a distinct form of braid word [2, 3]:
While knot 63 has three positive and three negative crossings, knot 83 and knots C3 with even C > 6 have C - 4 crossings of one sign and four crossings of the other sign [5] . 4. Concluding remarks Except for the 73 and 93 knots, the braid words are recorded here for the mirror image of Rolfsen [5] knot diagrams: our braid words will conform in crossing signs to Rolfsen diagrams if the sign of power exponents of the braid generators will be reversed, or the braid generators will be interpreted according to Akutsu et al. [3] . Expressions of braid words can be transformed according to the rules known for braid moves [1-4, 8-12]. The standardized expression of braid words for twist knots, (C with = 1, 2, 3, enables one to see the regular pattern of the words and thus to write the braid words of twist knots for general values of crossing number. The twist knots 4 1 and 62 are synthesized in organic stereochemistry. In laboratory syntheses of long organic molecular chains, in particular of polyether ladders, a "hook and ladder" approach to the synthesis of the twist knot 41 is dis- cussed in Ref. [20]. In the processive recombination of a supercoiled DNA ring [17], the successive products appear in a sequence with an increasing number of super- coil crossings [17, 21-24]. In the electron microscope, the supercoils of a DNA ring Braids for Twist Knots 3 ca e see e ocessi e ecom i a io o e A i gs y a eso ase e yme [17 1- ] yie s e wo-com o e i k ? e wis k o i e wo-com o e i k 5? a e wis k o e ai wo s o e wo-com o e i ks o ca e a es a e
A e iew [ ] o e mec a ism o A ecom i a io asse s io ogica y e wo mos im o a ami ies o k o s a ca e a es a e e o us a wis ami ies" a ese s e i us a i e e am es
e e e ces
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