K11n89: Difference between revisions

Latest revision as of 02:45, 3 September 2005

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n89 at Knotilus!
	[edit K11n89 Quick Notes]

[edit K11n89 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₄₂₅₁ X_10,3,11,4 X_12,6,13,5 X_7,16,8,17 X_9,18,10,19 X_2,11,3,12 X_20,13,21,14 X_15,6,16,7 X_17,8,18,9 X_22,19,1,20 X_14,21,15,22
Gauss code	1, -6, 2, -1, 3, 8, -4, 9, -5, -2, 6, -3, 7, -11, -8, 4, -9, 5, 10, -7, 11, -10
Dowker-Thistlethwaite code	4 10 12 -16 -18 2 20 -6 -8 22 14

A Braid Representative

A Morse Link Presentation

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11n89/ThurstonBennequinNumber
Hyperbolic Volume	12.7808
A-Polynomial	See Data:K11n89/A-polynomial

[edit Notes for K11n89's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	[math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant	4

[edit Notes for K11n89's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+7 t^2-14 t+17-14 t^{-1} +7 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6+z^4+5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 61, -4 }
Jones polynomial	[math]\displaystyle{ 2 q^{-2} -4 q^{-3} +7 q^{-4} -9 q^{-5} +11 q^{-6} -10 q^{-7} +8 q^{-8} -6 q^{-9} +3 q^{-10} - q^{-11} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^{10}-a^{10}+2 z^4 a^8+4 z^2 a^8+a^8-z^6 a^6-3 z^4 a^6-3 z^2 a^6-a^6+2 z^4 a^4+5 z^2 a^4+2 a^4 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-6 z^4 a^{12}+2 z^2 a^{12}+4 z^7 a^{11}-7 z^5 a^{11}+2 z^3 a^{11}-z a^{11}+3 z^8 a^{10}-3 z^6 a^{10}-2 z^2 a^{10}+a^{10}+z^9 a^9+4 z^7 a^9-9 z^5 a^9+5 z^3 a^9+5 z^8 a^8-10 z^6 a^8+11 z^4 a^8-4 z^2 a^8+a^8+z^9 a^7+z^7 a^7-2 z^3 a^7+2 z a^7+2 z^8 a^6-4 z^6 a^6+8 z^4 a^6-6 z^2 a^6+a^6+z^7 a^5+z^5 a^5-3 z^3 a^5+3 z^4 a^4-6 z^2 a^4+2 a^4 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{34}+q^{30}-2 q^{28}+q^{26}-q^{24}-q^{22}+2 q^{20}-q^{18}+3 q^{16}-q^{14}+2 q^{10}-q^8+2 q^6 }[/math]
The G2 invariant	Data:K11n89/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n89 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["K11n89"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= [math]\displaystyle{ -t^3+7 t^2-14 t+17-14 t^{-1} +7 t^{-2} - t^{-3} }[/math]

In[5]:= Conway[K][z]

Out[5]= [math]\displaystyle{ -z^6+z^4+5 z^2+1 }[/math]

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= [math]\displaystyle{ \{1\} }[/math]

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 61, -4 }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= [math]\displaystyle{ 2 q^{-2} -4 q^{-3} +7 q^{-4} -9 q^{-5} +11 q^{-6} -10 q^{-7} +8 q^{-8} -6 q^{-9} +3 q^{-10} - q^{-11} }[/math]

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= [math]\displaystyle{ -z^2 a^{10}-a^{10}+2 z^4 a^8+4 z^2 a^8+a^8-z^6 a^6-3 z^4 a^6-3 z^2 a^6-a^6+2 z^4 a^4+5 z^2 a^4+2 a^4 }[/math]

In[10]:= Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]= [math]\displaystyle{ z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-6 z^4 a^{12}+2 z^2 a^{12}+4 z^7 a^{11}-7 z^5 a^{11}+2 z^3 a^{11}-z a^{11}+3 z^8 a^{10}-3 z^6 a^{10}-2 z^2 a^{10}+a^{10}+z^9 a^9+4 z^7 a^9-9 z^5 a^9+5 z^3 a^9+5 z^8 a^8-10 z^6 a^8+11 z^4 a^8-4 z^2 a^8+a^8+z^9 a^7+z^7 a^7-2 z^3 a^7+2 z a^7+2 z^8 a^6-4 z^6 a^6+8 z^4 a^6-6 z^2 a^6+a^6+z^7 a^5+z^5 a^5-3 z^3 a^5+3 z^4 a^4-6 z^2 a^4+2 a^4 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

In[3]:= K = Knot["K11n89"];

In[4]:= {A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]= { [math]\displaystyle{ -t^3+7 t^2-14 t+17-14 t^{-1} +7 t^{-2} - t^{-3} }[/math], [math]\displaystyle{ 2 q^{-2} -4 q^{-3} +7 q^{-4} -9 q^{-5} +11 q^{-6} -10 q^{-7} +8 q^{-8} -6 q^{-9} +3 q^{-10} - q^{-11} }[/math] }

In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]= {}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {}

Vassiliev invariants

V₂ and V₃:

(5, -12)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

[math]\displaystyle{ 20 }[/math]

[math]\displaystyle{ -96 }[/math]

[math]\displaystyle{ 200 }[/math]

[math]\displaystyle{ \frac{1702}{3} }[/math]

[math]\displaystyle{ \frac{266}{3} }[/math]

[math]\displaystyle{ -1920 }[/math]

[math]\displaystyle{ -3680 }[/math]

[math]\displaystyle{ -640 }[/math]

[math]\displaystyle{ -512 }[/math]

[math]\displaystyle{ \frac{4000}{3} }[/math]

[math]\displaystyle{ 4608 }[/math]

[math]\displaystyle{ \frac{34040}{3} }[/math]

[math]\displaystyle{ \frac{5320}{3} }[/math]

[math]\displaystyle{ \frac{145759}{6} }[/math]

[math]\displaystyle{ 438 }[/math]

[math]\displaystyle{ \frac{87662}{9} }[/math]

[math]\displaystyle{ \frac{3397}{18} }[/math]

[math]\displaystyle{ \frac{7615}{6} }[/math]

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-4 is the signature of K11n89. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-3

2

-5

3

1

-2

-7

4

1

3

-9

5

3

-2

-11

6

4

2

-13

4

5

1

-15

4

6

-2

-17

2

4

2

-19

1

4

-3

-21

2

-23

1

-1

Integral Khovanov Homology

(db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math]	[math]\displaystyle{ i=-5 }[/math]	[math]\displaystyle{ i=-3 }[/math]
[math]\displaystyle{ r=-9 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-8 }[/math]	[math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-7 }[/math]	[math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math]	[math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-6 }[/math]	[math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math]	[math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-5 }[/math]	[math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math]	[math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-4 }[/math]	[math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math]	[math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=-3 }[/math]	[math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math]	[math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=-2 }[/math]	[math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math]	[math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-1 }[/math]	[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math]	[math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=0 }[/math]	[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z}^{2} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages.

See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

Back to the top.

K11n88

K11n90

@@ Line 16: / Line 16: @@
 k = 89 |
 KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-6,2,-1,3,8,-4,9,-5,-2,6,-3,7,-11,-8,4,-9,5,10,-7,11,-10/goTop.html |
-braid_table     = <table cellspacing=0 cellpadding=0 border=0>
+braid_table     = <table cellspacing=0 cellpadding=0 border=0 style="white-space: pre">
 <tr><td>[[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr>
 <tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]]</td></tr>
@@ Line 56: / Line 56: @@
          <td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
          </tr>
-         <tr valign=top><td colspan=2>Loading KnotTheory` (version of September 2, 2005, 15:8:39)...</td></tr>
+         <tr valign=top><td colspan=2>Loading KnotTheory` (version of September 3, 2005, 2:11:43)...</td></tr>
          <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Crossings[Knot[11, NonAlternating, 89]]</nowiki></pre></td></tr>
 <tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>11</nowiki></pre></td></tr>

K11n89: Difference between revisions

Latest revision as of 02:45, 3 September 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

Computer Talk

Modifying This Page

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