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An oriented link that has a diagram with all positive crossings (cf. also [[Link|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|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|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 $K$, $n &gt; 0$; cf. also [[Surgery|Surgery]]) is a homology $3$-sphere that does not bound a compact, smooth homology $4$-ball, [[#References|[a1]]], [[#References|[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====
<table><tr><td valign="top">[a1]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  P.B. Kronheimer,   T.S. Mrowka,   "Gauge theory for embedded surfaces. I"  ''Topology'' , '''32''' :  4  (1993)  pp. 773–826</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  W.W. Menasco,   "The Bennequin–Milnor unknotting conjectures"  ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''318''' :  9  (1994)  pp. 831–836</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  T. Nakamura,   "Four-genus and unknotting number of positive knots and links"  ''Osaka J. Math.'' , '''37'''  (2000)  pp. to appear</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  J.H. Przytycki,   "Positive knots have negative signature"  ''Bull. Acad. Polon. Math.'' , '''37'''  (1989)  pp. 559–562</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  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)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  L. Rudolph,   "Nontrivial positive braids have positive signature"  ''Topology'' , '''21''' :  3  (1982)  pp. 325–327</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  L. Rudolph,   "Quasipositvity as an obstruction to sliceness"  ''Bull. Amer. Math. Soc.'' , '''29'''  (1993)  pp. 51–59</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  L. Rudolph,   "Positive links are strongly quasipositive" , ''Proc. Kirbyfest'' , ''Geometry and Topology Monographs'' , '''2'''  (1999)  pp. 555–562</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  K. Taniyama,   "A partial order of knots"  ''Tokyo J. Math.'' , '''12''' :  1  (1989)  pp. 205–229</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  P. Traczyk,   "Nontrivial negative links have positive signature"  ''Manuscripta Math.'' , '''61''' :  3  (1988)  pp. 279–284</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  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</td></tr></table>