Positive link

An oriented link that has a diagram with all positive crossings (cf. also Link). More generally, a link is $m$-almost positive if it has a diagram with all but $m$ of its crossings being positive.

The unknotting number (Gordian number) of a positive link is equal to

\begin{equation*} \frac { 1 } { 2 } ( c ( D ) - s ( D ) + \operatorname { com } ( D ) ), \end{equation*}

where $D$ is a positive diagram of the link, $c ( D )$ is the number of crossings, $s ( D )$ is the number of Seifert circles of $D$, and $\operatorname { com }( D )$ is the number of components of the link (this generalizes the Milnor unknotting conjecture, 1969, and the Bennequin conjecture, 1981). Furthermore, for a positive knot the unknotting number is equal to the $4$-ball genus of the knot, to the genus of the knot (cf. also Knot theory), to the planar genus of the knot (from the Seifert construction), to the minimal degree of the Jones polynomial, and to half the degree of the Alexander polynomial.

One can define a relation $\geq$ on links by $L _ { 1 } \geq L _ { 2 }$ if and only if $L_{2}$ can be obtained from $L_1$ by changing some positive crossings of $L_1$. This relation allows one to express several fundamental properties of positive (and $m$-almost positive) links:

1) If $K$ is a positive knot, then $K \geq ( 5,2 )$ positive torus knot unless $K$ is a connected sum of pretzel knots $L ( p _ { 1 } , p _ { 2 } , p _ { 3 } )$, where $p _ { 1 }$, $p_2$ and $p_3$ are positive odd numbers;

a) if $K$ is a non-trivial positive knot, then either the signature $\sigma ( K ) \leq - 4$ or $K$ is a pretzel knot $L ( p _ { 1 } , p _ { 2 } , p _ { 3 } )$ (and then $\sigma ( K ) = - 2$);

b) if a positive knot has unknotting number one, then it is a positive twist knot.

2) Let $L$ be a non-trivial $1$-almost positive link. Then $L \geq $ right-handed trefoil knot (plus trivial components), or $L \geq $ right-handed Hopf link (plus trivial components). In particular, $L$ has a negative signature.

3) If $K$ is a $2$-almost positive knot, then either

i) $K \geq $ right handed trefoil; or

ii) $K \geq $ mirror image of the $6_2$-knot ($\sigma _ { 1 } ^ { 3 } \sigma _ { 2 } ^ { - 1 } \sigma _ { 1 } \sigma _ { 2 } ^ { - 1 }$ in the braid notation); or

iii) $K$ is a twist knot with a negative clasp.

4) If $K$ is a $2$-almost positive knot different from a twist knot with a negative clasp, then $K$ has negative signature and $K ( 1 / n )$ (i.e. $1 / n$ surgery on $K$, $n > 0$; cf. also Surgery) is a homology $3$-sphere that does not bound a compact, smooth homology $4$-ball, [a1], [a6];

5) if $K$ is a non-trivial $2$-almost positive knot different from the Stevedore knot, then $K$ is not a slice knot;

6) if $K$ is a non-trivial $2$-almost positive knot different from the figure eight knot, then $K$ is not amphicheiral.

7) Let $K$ be a $3$-almost positive knot. Then either $K \geq $ trivial knot or $K$ is the left-handed trefoil knot (plus positive knots as connected summands). In particular, either $K$ has a non-positive signature or $K$ is the left-handed trefoil knot.

References

[a1]	T. Cochran, E. Gompf, "Applications of Donaldson's theorems to classical knot concordance, homology $3$-spheres and property $P$" Topology , 27 : 4 (1988) pp. 495–512
[a2]	P.B. Kronheimer, T.S. Mrowka, "Gauge theory for embedded surfaces. I" Topology , 32 : 4 (1993) pp. 773–826
[a3]	W.W. Menasco, "The Bennequin–Milnor unknotting conjectures" C.R. Acad. Sci. Paris Sér. I Math. , 318 : 9 (1994) pp. 831–836
[a4]	T. Nakamura, "Four-genus and unknotting number of positive knots and links" Osaka J. Math. , 37 (2000) pp. to appear
[a5]	J.H. Przytycki, "Positive knots have negative signature" Bull. Acad. Polon. Math. , 37 (1989) pp. 559–562
[a6]	J.H. Przytycki, K. Taniyama, "Almost positive links have negative signature" preprint (1991) (See: Abstracts Amer. Math. Soc., June 1991, Issue 75, Vol. 12 (3), p. 327, $*$91T-57-69)
[a7]	L. Rudolph, "Nontrivial positive braids have positive signature" Topology , 21 : 3 (1982) pp. 325–327
[a8]	L. Rudolph, "Quasipositvity as an obstruction to sliceness" Bull. Amer. Math. Soc. , 29 (1993) pp. 51–59
[a9]	L. Rudolph, "Positive links are strongly quasipositive" , Proc. Kirbyfest , Geometry and Topology Monographs , 2 (1999) pp. 555–562
[a10]	K. Taniyama, "A partial order of knots" Tokyo J. Math. , 12 : 1 (1989) pp. 205–229
[a11]	P. Traczyk, "Nontrivial negative links have positive signature" Manuscripta Math. , 61 : 3 (1988) pp. 279–284
[a12]	J.M. van Buskirk, "Positive knots have positive Conway polynomials" , Knot Theory And Manifolds (Vancouver, B.C., 1983) , Lecture Notes in Mathematics , 1144 , Springer (1985) pp. 146–159