T(4,3) Quick Notes

2011-04-06T05:55:48Z

Dthurston:

T(4,3) Quick Notes

2011-04-06T05:55:33Z

Dthurston: New page: See also 8_19

L9n15 Quick Notes

2009-09-04T01:52:39Z

Dthurston: New page: This is a Seifert-fibered link: it is the trefoil plus the core of one solid torus.

This is a Seifert-fibered link: it is the trefoil plus the core of one solid torus.

L9n15 Further Notes and Views

2009-09-04T01:50:27Z

Dthurston:

{{Knot View Template|
image = Triune.jpg|
text = The Triune sculpture in Philadelpdia by Robert Engman, showing a Seifert surface for L9n15. Photo by [http://www.flickr.com/photos/29923994@N03/3105524757/in/photostream/ sameold2008]}}

L9n15 Further Notes and Views

2009-09-04T01:49:57Z

Dthurston: New page: {{Knot View Template| image = Triune.jpg| text = The Triune sculpture in Philadelpdia by Robert Engman, showing a Seifert surface for L9n15. Picture by [http://www.flickr.com/photos/2992...

{{Knot View Template|
image = Triune.jpg|
text = The Triune sculpture in Philadelpdia by Robert Engman, showing a Seifert surface for L9n15. Picture by [http://www.flickr.com/photos/29923994@N03/3105524757/in/photostream/ sameold2008]}}

File:Triune.jpg

2009-09-04T01:47:14Z

Dthurston: Triune sculpture

Triune sculpture

Heegaard Floer Knot Homology

2008-03-07T23:34:57Z

Dthurston:

{{Manual TOC Sidebar}}

In 2007, [http://www.math.unizh.ch/user/jdroz/ Jean-Marie Droz] of the University of Zurich (working along with [http://www.math.unizh.ch/index.php?id=1819&no_cache=1&key1=578&no_cache=1 Anna Beliakova]) wrote a Python program to compute the (hat-version) Heegaard-Floer Knot Homology <math>\widehat{\operatorname{HFK}}(K)</math> of a knot <math>K</math>. His program is integrated into <code>KnotTheory`</code>, though to run it, you must have [http://python.org/ Python] as well as the Python library [http://psyco.sourceforge.net/ Psyco] installed on your system.

{{Startup Note}}



{{HelpAndAbout|
n = 1 |
n1 = 2 |
in = <nowiki>HFKHat</nowiki> |
out= <nowiki>HFKHat[K][t,m] returns the Poincare polynomial of the Heegaard-Floer Knot Homology (hat version) of the knot K, in the Alexander variable t and the Maslov variable m.</nowiki> |
about= <nowiki>The Heegaard-Floer Knot Homology program was written by Jean-Marie Droz in 2007 at the University of Zurich, based on methods of Anna Beliakova's arXiv:07050669.</nowiki>}}


{| align=center
|[[Image:8_19.gif|thumb|180px|<center>[[8_19]]</center>]]
|[[Image:8_19_AP.gif|thumb|none|<center>in [[Arc Presentations|Arc Presentation]]</center>|180px]]
|}

The Heegaard-Floer Knot Homology is a categorification of the [[The Alexander-Conway Polynomial|Alexander polynomial]]. Let us test that for the knot [[8_19]]:



{{InOut|
n = 3 |
in = <nowiki>hfk = HFKHat[K = Knot[8, 19]][t, m]</nowiki> |
out= <nowiki> 2 -3 m 5 2 6 3
m + t + -- + m t + m t
2
t</nowiki>}}




{{InOut|
n = 4 |
in = <nowiki>{hfk /. m -> -1, Alexander[K][t]}</nowiki> |
out= <nowiki> -3 -2 2 3 -3 -2 2 3
{1 + t - t - t + t , 1 + t - t - t + t }</nowiki>}}


The knot [[8_19]] is the first knot in the [[The Rolfsen Knot Table|Rolfsen Knot Table]] whose Heegaard-Floer Knot Homology is not "diagonal". Let us test that. The homology <math>\widehat{\operatorname{HFK}}(K)</math> is "on diagonal", iff its Poincare polynomial, evaluated at <math>m=1/t</math>, is a monomial:



{{InOut|
n = 5 |
in = <nowiki>Select[AllKnots[{3, 8}], (Head[HFKHat[#][t, 1/t]] == Plus) &]</nowiki> |
out= <nowiki>{Knot[8, 19]}</nowiki>}}




{{InOut|
n = 6 |
in = <nowiki>hfk /. m -> 1/t</nowiki> |
out= <nowiki>4 -2
-- + t
3
t</nowiki>}}


{{Knot Image Pair|K11n34|gif|K11n42|gif}}

The (mirrored) Conway knot [[K11n34]] and the (mirrored) Kinoshita-Terasaka knot [[K11n42]] are a mutant pair, and are notoriously difficult to tell apart. Let us check that an array of standard knot polynomials fails to separate them, yet <math>\widehat{\operatorname{HFK}}</math> succeeds:



{{InOut|
n = 7 |
in = <nowiki>K1 = Knot["K11n34"]; K2 = Knot["K11n42"];
test[invt_] := (invt[K1] =!= invt[K2]);
test /@ {
Alexander, MultivariableAlexander, Jones, HOMFLYPT, Kauffman, Kh, HFKHat
}</nowiki> |
out= <nowiki>{False, False, False, False, False, False, True}</nowiki>}}


Indeed,



{{InOut|
n = 8 |
in = <nowiki>{HFKHat[K1][t, m], HFKHat[K2][t, m]}</nowiki> |
out= <nowiki> 2 1 1 3 3 3 3
{3 + - + ----- + ----- + ----- + ----- + ---- + --- + 3 t + 3 m t +
m 4 3 3 3 3 2 2 2 2 m t
m t m t m t m t m t

2 2 2 2 3 3 3
3 m t + 3 m t + m t + m t ,

6 1 1 4 4 2 2 2
7 + - + ----- + ----- + ---- + --- + 4 t + 4 m t + m t + m t }
m 3 2 2 2 2 m t
m t m t m t</nowiki>}}


On July 6, 2006, [[User:AnonMoos]] [[User_talk:Drorbn#Here.27s_one|asked]] [[User:Drorbn]] if he could identify the knot in the left hand side picture below. At the time it was impossible using the tools available with <code>KnotTheory`</code> - using any of many invariants, the answer can be found to be either the mirror of [[K11n34]] or the mirror of [[K11n42]], but <code>KnotTheory`</code> couldn't tell which one it is (though of course, it is possible to do it "by hand"). The 2007 addition <math>\widehat{\operatorname{HFK}}</math> does the job, though. Indeed, we first extract the mystery knot's [[DT (Dowker-Thistlethwaite) Codes|DT (Dowker-Thistlethwaite) Code]] using the picture on the right hand side below, then compute <math>\widehat{\operatorname{HFK}}</math>, and then search for it within the <math>\widehat{\operatorname{HFK}}</math>'s of all knots with up to 11 crossings:

{| align=center width=80%
|align=center|[[Image:Gateknot.jpg|240px]]
|align=center|[[Image:Gateknot DT Labeled.png|252px]]
|}



{{In|
n = 9 |
in = <nowiki>K3 = DTCode[6, 8, 14, 12, 4, -18, 2, -20, -22, -10, -16];</nowiki>}}




{{InOut|
n = 10 |
in = <nowiki>H = HFKHat[Mirror[K3]][t, m]</nowiki> |
out= <nowiki> 2 1 1 3 3 3 3
3 + - + ----- + ----- + ----- + ----- + ---- + --- + 3 t + 3 m t +
m 4 3 3 3 3 2 2 2 2 m t
m t m t m t m t m t

2 2 2 2 3 3 3
3 m t + 3 m t + m t + m t</nowiki>}}




{{InOut|
n = 11 |
in = <nowiki>Select[AllKnots[], HFKHat[#][t, m] == H &]</nowiki> |
out= <nowiki>{Knot[11, NonAlternating, 34]}</nowiki>}}


And so the mystery knot is the Conway knot, the mirror of [[K11n34]].

Knot Atlas - User contributions [en]

T(4,3) Quick Notes

T(4,3) Quick Notes

L9n15 Quick Notes

L9n15 Further Notes and Views

L9n15 Further Notes and Views

File:Triune.jpg

Heegaard Floer Knot Homology