Khovanov Homology - Revision history

ScottDataRobot at 22:23, 21 February 2013

2013-02-21T22:23:39Z

← Older revision		Revision as of 18:23, 21 February 2013
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	n = 2 \|		n = 2 \|
	in = <nowiki>Kh</nowiki> \|		in = <nowiki>Kh</nowiki> \|
	out= <nowiki>Kh[L][q, t] returns the Poincare polynomial of the Khovanov Homology of a knot/link L (over a field of characteristic 0) in terms of the variables q and t. Kh[L, Program -> prog] uses the program prog to perform the computation. The currently available programs are "FastKh", written in Mathematica by Dror Bar-Natan in the winter of 2005 and "JavaKh" (default), written in java (java 1.5 required!) by Jeremy Green in the summer of 2005. The java program is several thousand times faster than the Mathematica program, though java may not be available on some systems. "JavaKh" also takes the option "Modulus -> p" which changes the characteristic of the ground field to p. If p==0 JavaKh works over the rational numbers; if p==Null JavaKh works over Z (see ?ZMod for the output format).</nowiki>}}		out= <nowiki>Kh[L][q, t] returns the Poincare polynomial of the Khovanov Homology of a knot/link L (over a field of characteristic 0) in terms of the variables q and t. Kh[L, Program -> prog] uses the
			program prog to perform the computation. The currently available programs are "FastKh", written in Mathematica by Dror Bar-Natan in the winter of 2005, "JavaKh-v1", written in java (java 1.5
			required!) by Jeremy Green in the summer of 2005 and "JavaKh-v2" (default), an update of "JavaKh-v1" (now requiring java 1.6) written by Scott Morrison in 2008.
			("JavaKh" is also available, currently an alias for "JavaKh-v2".)
			The java programs are several thousand times faster than the Mathematica program, though java may not be available on some systems. "JavaKh2" also takes the option "Modulus -> p" which changes the characteristic of the ground field to p. If p==0 JavaKh works over the rational numbers; if p==Null JavaKh works over Z (see ?ZMod for the output format).</nowiki>}}
	<!--END-->		<!--END-->

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	in = <nowiki>Options[Kh]</nowiki> \|		in = <nowiki>Options[Kh]</nowiki> \|
	out= <nowiki>{ExpansionOrder -> Automatic, Program -> JavaKh, Modulus -> 0,		out= <nowiki>{ExpansionOrder -> Automatic, Program -> JavaKh-v2, Modulus -> 0,

	JavaOptions -> }</nowiki>}}		KnotTheory`FastKh`Universal -> False, JavaOptions -> }</nowiki>}}
	<!--END-->		<!--END-->

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	n = 11 \|		n = 11 \|
	in = <nowiki>JavaOptions</nowiki> \|		in = <nowiki>JavaOptions</nowiki> \|
	out= <nowiki>JavaOptions is an option to Kh. Kh[L, Program -> "~~JavaKh~~", JavaOptions -> jopts] calls java with options jopts. Thus for example, JavaOptions -> "-Xmx256m" sets the maximum java heap size to 256MB - useful for large computations.</nowiki>}}		out= <nowiki>JavaOptions is an option to Kh. Kh[L, Program -> "JavaKh2", JavaOptions -> jopts] calls java with options jopts. Thus for example, JavaOptions -> "-Xmx256m" sets the maximum java heap size to 256MB - useful for large computations.</nowiki>}}
	<!--END-->		<!--END-->

Scott at 23:04, 22 May 2012

2012-05-22T23:04:52Z

← Older revision		Revision as of 19:04, 22 May 2012
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	(Type <tt>java JavaKh -help</tt> for some further help).		(Type <tt>java JavaKh -help</tt> for some further help).

	('''Warning''', as of August 2008, you need to include the .jar files in JavaKh/jars on the classpath. If this causes confusion, ask Scott for a script that manages this automatically, or look at p. 29 of [http://arxiv.org/~~pdf~~/0906.5177] for an example.)		('''Warning''', as of August 2008, you need to include the .jar files in JavaKh/jars on the classpath. If this causes confusion, ask Scott for a script that manages this automatically, or look at p. 29 of [http://arxiv.org/abs/0906.5177] for an example.)

	===Universal Khovanov homology, and reduced homology===		===Universal Khovanov homology, and reduced homology===

Scott at 23:04, 22 May 2012

2012-05-22T23:04:31Z

← Older revision		Revision as of 19:04, 22 May 2012
Line 263:		Line 263:
	(Type <tt>java JavaKh -help</tt> for some further help).		(Type <tt>java JavaKh -help</tt> for some further help).

	('''Warning''', as of August 2008, you need to include the .jar files in JavaKh/jars on the classpath~~, because of the changes [http://katlas.org/fisheye/changelog/KnotTheory/?cs=292 described here]~~. If this causes confusion, ask Scott for a script that manages this automatically.)		('''Warning''', as of August 2008, you need to include the .jar files in JavaKh/jars on the classpath. If this causes confusion, ask Scott for a script that manages this automatically, or look at p. 29 of [http://arxiv.org/pdf/0906.5177] for an example.)

	===Universal Khovanov homology, and reduced homology===		===Universal Khovanov homology, and reduced homology===

Scott: /* Universal Khovanov homology, and reduced homology */

2010-06-26T21:07:02Z

Universal Khovanov homology, and reduced homology

← Older revision		Revision as of 17:07, 26 June 2010
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	Unfortunately, much of the mathematics behind <code>UniversalKh</code> is not in print. There's some explanation in [http://tqft.net/kyoto2 Scott's slides] from Kyoto.		Unfortunately, much of the mathematics behind <code>UniversalKh</code> is not in print. There's some explanation in [http://tqft.net/kyoto2 Scott's slides] from Kyoto.

			UniversalKh also takes the same JavaOptions option described above for Kh, which may be necessary to allow enough memory for large computations.

	===References===		===References===

Scott: /* Universal Khovanov homology, and reduced homology */

2010-06-26T21:06:10Z

Universal Khovanov homology, and reduced homology

@@ Line 266: / Line 266: @@
 ===Universal Khovanov homology, and reduced homology===
-The KnotTheory` package can now compute the universal homology over Q, and reduced homology over Q. Type <code>?UniversalKh</code> or <code>?ReducedKh</code> for more information. The output of <code>UniversalKh</code> is a <math>\mathbb{Z}[q,t]</math>-linear combination of symbols <code>KhE</code> and <code>KhC[n]</code> for positive integer <math>n</math>. <code>KhE</code> corresponds to an exceptional pair, <code>KhC[1]</code> to a knight's move, and so on. See [http://tqft.net/kyoto2 Scott's slides] for further explanations.
+The KnotTheory` package can now compute the universal homology over <math>\mathbb{Q}</math>, the reduced homology over <math>\mathbb{Q}</math> as well as directly computing the s-invariant. Type <code>?UniversalKh</code>, <code>?KhReduced</code> or <code>?sInvariant</code> for more information.
 ===References===

Drorbn: Reverted edits by RaccoAcels (Talk); changed back to last version by Scott

2008-12-17T11:45:50Z

Reverted edits by RaccoAcels (Talk); changed back to last version by Scott

← Older revision		Revision as of 07:45, 17 December 2008
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	erlitrda
	{{Manual TOC Sidebar}}		{{Manual TOC Sidebar}}

	(See also: [[Tweaking JavaKh]])		(See also: [[Tweaking JavaKh]])

	The Khovanov Homology <math>\operatorname{\it KH}(L)</math> of a knot or a link <math>L</math>, also known as Khovanov's categorification of the Jones polynomial of <math>L</math>, was defined by Khovanov in {{ref\|Khovanov1}} (also check {{ref\|Bar-Natan1}}), where the notation is closer to the notation used here). It is a graded homology theory; each homology group <math>\operatorname{\it KH}^r(L)</math> is in itself a direct sum <math>\bigoplus_j\operatorname{\it KH}^r_j(L)</math> of homogeneous components. Over a field one can form the two-variable "~~PoincarÃ©~~ polynomial" <math>\operatorname{\it Kh}(L)</math> (which deserves the name "the Khovanov polynomial of <math>L</math>"),		The Khovanov Homology <math>\operatorname{\it KH}(L)</math> of a knot or a link <math>L</math>, also known as Khovanov's categorification of the Jones polynomial of <math>L</math>, was defined by Khovanov in {{ref\|Khovanov1}} (also check {{ref\|Bar-Natan1}}), where the notation is closer to the notation used here). It is a graded homology theory; each homology group <math>\operatorname{\it KH}^r(L)</math> is in itself a direct sum <math>\bigoplus_j\operatorname{\it KH}^r_j(L)</math> of homogeneous components. Over a field one can form the two-variable "Poincaré polynomial" <math>\operatorname{\it Kh}(L)</math> (which deserves the name "the Khovanov polynomial of <math>L</math>"),

	<center><math>\operatorname{\it Kh}(L)(q,t) := \sum_{r,j}t^rq^j\dim\operatorname{\it KH}^r_j(L)</math>.</center>		<center><math>\operatorname{\it Kh}(L)(q,t) := \sum_{r,j}t^rq^j\dim\operatorname{\it KH}^r_j(L)</math>.</center>

RaccoAcels: trcolio

2008-12-16T17:45:07Z

trcolio

← Older revision		Revision as of 13:45, 16 December 2008
Line 1:		Line 1:
			erlitrda
	{{Manual TOC Sidebar}}		{{Manual TOC Sidebar}}

	(See also: [[Tweaking JavaKh]])		(See also: [[Tweaking JavaKh]])

	The Khovanov Homology <math>\operatorname{\it KH}(L)</math> of a knot or a link <math>L</math>, also known as Khovanov's categorification of the Jones polynomial of <math>L</math>, was defined by Khovanov in {{ref\|Khovanov1}} (also check {{ref\|Bar-Natan1}}), where the notation is closer to the notation used here). It is a graded homology theory; each homology group <math>\operatorname{\it KH}^r(L)</math> is in itself a direct sum <math>\bigoplus_j\operatorname{\it KH}^r_j(L)</math> of homogeneous components. Over a field one can form the two-variable "~~Poincaré~~ polynomial" <math>\operatorname{\it Kh}(L)</math> (which deserves the name "the Khovanov polynomial of <math>L</math>"),		The Khovanov Homology <math>\operatorname{\it KH}(L)</math> of a knot or a link <math>L</math>, also known as Khovanov's categorification of the Jones polynomial of <math>L</math>, was defined by Khovanov in {{ref\|Khovanov1}} (also check {{ref\|Bar-Natan1}}), where the notation is closer to the notation used here). It is a graded homology theory; each homology group <math>\operatorname{\it KH}^r(L)</math> is in itself a direct sum <math>\bigoplus_j\operatorname{\it KH}^r_j(L)</math> of homogeneous components. Over a field one can form the two-variable "PoincarÃ© polynomial" <math>\operatorname{\it Kh}(L)</math> (which deserves the name "the Khovanov polynomial of <math>L</math>"),

	<center><math>\operatorname{\it Kh}(L)(q,t) := \sum_{r,j}t^rq^j\dim\operatorname{\it KH}^r_j(L)</math>.</center>		<center><math>\operatorname{\it Kh}(L)(q,t) := \sum_{r,j}t^rq^j\dim\operatorname{\it KH}^r_j(L)</math>.</center>

Scott at 21:07, 3 August 2008

2008-08-03T21:07:55Z

← Older revision		Revision as of 17:07, 3 August 2008
Line 262:		Line 262:

	(Type <tt>java JavaKh -help</tt> for some further help).		(Type <tt>java JavaKh -help</tt> for some further help).

			('''Warning''', as of August 2008, you need to include the .jar files in JavaKh/jars on the classpath, because of the changes [http://katlas.org/fisheye/changelog/KnotTheory/?cs=292 described here]. If this causes confusion, ask Scott for a script that manages this automatically.)

	===Universal Khovanov homology, and reduced homology===		===Universal Khovanov homology, and reduced homology===

Drorbn: Reverted edits by CnadrOnacc (Talk); changed back to last version by Drorbn

2008-07-12T18:11:18Z

Reverted edits by CnadrOnacc (Talk); changed back to last version by Drorbn

← Older revision		Revision as of 14:11, 12 July 2008
Line 1:		Line 1:
	orc4tnor
	{{Manual TOC Sidebar}}		{{Manual TOC Sidebar}}

	(See also: [[Tweaking JavaKh]])		(See also: [[Tweaking JavaKh]])

	The Khovanov Homology <math>\operatorname{\it KH}(L)</math> of a knot or a link <math>L</math>, also known as Khovanov's categorification of the Jones polynomial of <math>L</math>, was defined by Khovanov in {{ref\|Khovanov1}} (also check {{ref\|Bar-Natan1}}), where the notation is closer to the notation used here). It is a graded homology theory; each homology group <math>\operatorname{\it KH}^r(L)</math> is in itself a direct sum <math>\bigoplus_j\operatorname{\it KH}^r_j(L)</math> of homogeneous components. Over a field one can form the two-variable "~~PoincarÃ©~~ polynomial" <math>\operatorname{\it Kh}(L)</math> (which deserves the name "the Khovanov polynomial of <math>L</math>"),		The Khovanov Homology <math>\operatorname{\it KH}(L)</math> of a knot or a link <math>L</math>, also known as Khovanov's categorification of the Jones polynomial of <math>L</math>, was defined by Khovanov in {{ref\|Khovanov1}} (also check {{ref\|Bar-Natan1}}), where the notation is closer to the notation used here). It is a graded homology theory; each homology group <math>\operatorname{\it KH}^r(L)</math> is in itself a direct sum <math>\bigoplus_j\operatorname{\it KH}^r_j(L)</math> of homogeneous components. Over a field one can form the two-variable "Poincaré polynomial" <math>\operatorname{\it Kh}(L)</math> (which deserves the name "the Khovanov polynomial of <math>L</math>"),

	<center><math>\operatorname{\it Kh}(L)(q,t) := \sum_{r,j}t^rq^j\dim\operatorname{\it KH}^r_j(L)</math>.</center>		<center><math>\operatorname{\it Kh}(L)(q,t) := \sum_{r,j}t^rq^j\dim\operatorname{\it KH}^r_j(L)</math>.</center>

CnadrOnacc: racoupasvile

2008-07-12T15:26:11Z

racoupasvile

← Older revision		Revision as of 11:26, 12 July 2008
Line 1:		Line 1:
			orc4tnor
	{{Manual TOC Sidebar}}		{{Manual TOC Sidebar}}

	(See also: [[Tweaking JavaKh]])		(See also: [[Tweaking JavaKh]])

	The Khovanov Homology <math>\operatorname{\it KH}(L)</math> of a knot or a link <math>L</math>, also known as Khovanov's categorification of the Jones polynomial of <math>L</math>, was defined by Khovanov in {{ref\|Khovanov1}} (also check {{ref\|Bar-Natan1}}), where the notation is closer to the notation used here). It is a graded homology theory; each homology group <math>\operatorname{\it KH}^r(L)</math> is in itself a direct sum <math>\bigoplus_j\operatorname{\it KH}^r_j(L)</math> of homogeneous components. Over a field one can form the two-variable "~~Poincaré~~ polynomial" <math>\operatorname{\it Kh}(L)</math> (which deserves the name "the Khovanov polynomial of <math>L</math>"),		The Khovanov Homology <math>\operatorname{\it KH}(L)</math> of a knot or a link <math>L</math>, also known as Khovanov's categorification of the Jones polynomial of <math>L</math>, was defined by Khovanov in {{ref\|Khovanov1}} (also check {{ref\|Bar-Natan1}}), where the notation is closer to the notation used here). It is a graded homology theory; each homology group <math>\operatorname{\it KH}^r(L)</math> is in itself a direct sum <math>\bigoplus_j\operatorname{\it KH}^r_j(L)</math> of homogeneous components. Over a field one can form the two-variable "PoincarÃ© polynomial" <math>\operatorname{\it Kh}(L)</math> (which deserves the name "the Khovanov polynomial of <math>L</math>"),

	<center><math>\operatorname{\it Kh}(L)(q,t) := \sum_{r,j}t^rq^j\dim\operatorname{\it KH}^r_j(L)</math>.</center>		<center><math>\operatorname{\it Kh}(L)(q,t) := \sum_{r,j}t^rq^j\dim\operatorname{\it KH}^r_j(L)</math>.</center>