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An oriented link that has a diagram with all positive crossings (cf. also [[Link|Link]]). More generally, a link is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p1301202.png" />-almost positive if it has a diagram with all but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p1301203.png" /> of its crossings being positive.
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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
 
The unknotting number (Gordian number) of a positive link is equal to
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p1301204.png" /></td> </tr></table>
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\begin{equation*} \frac { 1 } { 2 } ( c ( D ) - s ( D ) + \operatorname { com } ( D ) ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p1301205.png" /> is a positive diagram of the link, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p1301206.png" /> is the number of crossings, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p1301207.png" /> is the number of Seifert circles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p1301208.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p1301209.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012011.png" />-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.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012012.png" /> on links by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012013.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012014.png" /> can be obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012015.png" /> by changing some positive crossings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012016.png" />. This relation allows one to express several fundamental properties of positive (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012017.png" />-almost positive) links:
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012018.png" /> is a positive knot, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012019.png" /> positive torus knot unless <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012021.png" /> is a connected sum of pretzel knots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012025.png" /> are positive odd numbers;
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012026.png" /> is a non-trivial positive knot, then either the signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012027.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012028.png" /> is a pretzel knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012029.png" /> (and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012030.png" />);
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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.
 
b) if a positive knot has unknotting number one, then it is a positive twist knot.
  
2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012031.png" /> be a non-trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012032.png" />-almost positive link. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012033.png" /> right-handed trefoil knot (plus trivial components), or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012034.png" /> right-handed Hopf link (plus trivial components). In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012035.png" /> has a negative signature.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012036.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012037.png" />-almost positive knot, then either
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3) If $K$ is a $2$-almost positive knot, then either
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012038.png" /> right handed trefoil; or
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i) $K \geq $ right handed trefoil; or
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012039.png" /> mirror image of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012041.png" />-knot (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012042.png" /> in the braid notation); or
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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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012043.png" /> is a twist knot with a negative clasp.
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iii) $K$ is a twist knot with a negative clasp.
  
4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012044.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012045.png" />-almost positive knot different from a twist knot with a negative clasp, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012046.png" /> has negative signature and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012047.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012049.png" /> surgery on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012051.png" />; cf. also [[Surgery|Surgery]]) is a homology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012052.png" />-sphere that does not bound a compact, smooth homology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012053.png" />-ball, [[#References|[a1]]], [[#References|[a6]]];
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012054.png" /> is a non-trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012055.png" />-almost positive knot different from the Stevedore knot, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012056.png" /> is not a slice knot;
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012057.png" /> is a non-trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012058.png" />-almost positive knot different from the figure eight knot, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012059.png" /> is not amphicheiral.
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6) if $K$ is a non-trivial $2$-almost positive knot different from the figure eight knot, then $K$ is not amphicheiral.
  
7) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012060.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012061.png" />-almost positive knot. Then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012062.png" /> trivial knot or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012063.png" /> is the left-handed trefoil knot (plus positive knots as connected summands). In particular, either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012064.png" /> has a non-positive signature or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012065.png" /> is the left-handed trefoil knot.
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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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012066.png" />-spheres and property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012067.png" />"  ''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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130120/p13012068.png" />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>
+
<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>

Latest revision as of 16:58, 1 July 2020

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
How to Cite This Entry:
Positive link. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive_link&oldid=50271
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article