K11a81: Difference between revisions

Latest revision as of 02:41, 3 September 2005

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a81 at Knotilus!
	[edit K11a81 Quick Notes]

[edit K11a81 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₄₂₅₁ X_10,3,11,4 X_12,6,13,5 X_14,7,15,8 X_22,10,1,9 X_2,11,3,12 X_18,13,19,14 X_20,16,21,15 X_8,17,9,18 X_6,20,7,19 X_16,22,17,21
Gauss code	1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, -8, 11, -5
Dowker-Thistlethwaite code	4 10 12 14 22 2 18 20 8 6 16

A Braid Representative

A Morse Link Presentation

Three dimensional invariants

Symmetry type	Chiral
Unknotting number	$\{1,2\}$
3-genus	4
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a81/ThurstonBennequinNumber
Hyperbolic Volume	16.051
A-Polynomial	See Data:K11a81/A-polynomial

[edit Notes for K11a81's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$4$
Rasmussen s-Invariant	-2

[edit Notes for K11a81's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-16t^{2}+26t-29+26t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 127, 2 }
Jones polynomial	$q^{7}-4q^{6}+8q^{5}-13q^{4}+18q^{3}-20q^{2}+20q-17+13q^{-1}-8q^{-2}+4q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-9z^{4}a^{-2}+3z^{4}a^{-4}+7z^{4}-2a^{2}z^{2}-6z^{2}a^{-2}+2z^{2}a^{-4}+6z^{2}+1$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+5az^{9}+12z^{9}a^{-1}+7z^{9}a^{-3}+4a^{2}z^{8}+15z^{8}a^{-2}+11z^{8}a^{-4}+8z^{8}+a^{3}z^{7}-13az^{7}-26z^{7}a^{-1}-z^{7}a^{-3}+11z^{7}a^{-5}-14a^{2}z^{6}-50z^{6}a^{-2}-14z^{6}a^{-4}+8z^{6}a^{-6}-42z^{6}-3a^{3}z^{5}+3az^{5}-23z^{5}a^{-3}-13z^{5}a^{-5}+4z^{5}a^{-7}+15a^{2}z^{4}+39z^{4}a^{-2}+2z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+44z^{4}+3a^{3}z^{3}+9az^{3}+18z^{3}a^{-1}+19z^{3}a^{-3}+5z^{3}a^{-5}-2z^{3}a^{-7}-5a^{2}z^{2}-9z^{2}a^{-2}+z^{2}a^{-4}+2z^{2}a^{-6}-13z^{2}-a^{3}z-4az-6za^{-1}-4za^{-3}-za^{-5}+1$
The A2 invariant	$-q^{12}+q^{10}+q^{8}-q^{6}+3q^{4}-3q^{2}+1+q^{-2}-2q^{-4}+5q^{-6}-3q^{-8}+3q^{-10}-q^{-12}-2q^{-14}+2q^{-16}-2q^{-18}+q^{-20}$
The G2 invariant	$q^{60}-3q^{58}+9q^{56}-19q^{54}+28q^{52}-32q^{50}+15q^{48}+29q^{46}-92q^{44}+157q^{42}-186q^{40}+140q^{38}-15q^{36}-174q^{34}+360q^{32}-449q^{30}+392q^{28}-169q^{26}-152q^{24}+452q^{22}-607q^{20}+545q^{18}-277q^{16}-87q^{14}+395q^{12}-526q^{10}+425q^{8}-144q^{6}-183q^{4}+416q^{2}-444+250q^{-2}+81q^{-4}-420q^{-6}+618q^{-8}-586q^{-10}+326q^{-12}+84q^{-14}-495q^{-16}+758q^{-18}-764q^{-20}+514q^{-22}-89q^{-24}-345q^{-26}+628q^{-28}-662q^{-30}+448q^{-32}-89q^{-34}-249q^{-36}+431q^{-38}-389q^{-40}+158q^{-42}+136q^{-44}-356q^{-46}+405q^{-48}-270q^{-50}+10q^{-52}+254q^{-54}-425q^{-56}+448q^{-58}-321q^{-60}+111q^{-62}+110q^{-64}-276q^{-66}+338q^{-68}-310q^{-70}+215q^{-72}-84q^{-74}-35q^{-76}+125q^{-78}-169q^{-80}+165q^{-82}-127q^{-84}+72q^{-86}-16q^{-88}-29q^{-90}+53q^{-92}-62q^{-94}+51q^{-96}-31q^{-98}+15q^{-100}+q^{-102}-8q^{-104}+10q^{-106}-10q^{-108}+6q^{-110}-3q^{-112}+q^{-114}$

Further Quantum Invariants

K11a81 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["K11a81"];

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

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

Out[4]= $-t^{4}+6t^{3}-16t^{2}+26t-29+26t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$

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

Out[5]= $-z^{8}-2z^{6}+1$

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]= $\{1\}$

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

Out[7]= { 127, 2 }

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

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

Out[8]= $q^{7}-4q^{6}+8q^{5}-13q^{4}+18q^{3}-20q^{2}+20q-17+13q^{-1}-8q^{-2}+4q^{-3}-q^{-4}$

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

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

Out[9]= $-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-9z^{4}a^{-2}+3z^{4}a^{-4}+7z^{4}-2a^{2}z^{2}-6z^{2}a^{-2}+2z^{2}a^{-4}+6z^{2}+1$

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

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

Out[10]= $2z^{10}a^{-2}+2z^{10}+5az^{9}+12z^{9}a^{-1}+7z^{9}a^{-3}+4a^{2}z^{8}+15z^{8}a^{-2}+11z^{8}a^{-4}+8z^{8}+a^{3}z^{7}-13az^{7}-26z^{7}a^{-1}-z^{7}a^{-3}+11z^{7}a^{-5}-14a^{2}z^{6}-50z^{6}a^{-2}-14z^{6}a^{-4}+8z^{6}a^{-6}-42z^{6}-3a^{3}z^{5}+3az^{5}-23z^{5}a^{-3}-13z^{5}a^{-5}+4z^{5}a^{-7}+15a^{2}z^{4}+39z^{4}a^{-2}+2z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+44z^{4}+3a^{3}z^{3}+9az^{3}+18z^{3}a^{-1}+19z^{3}a^{-3}+5z^{3}a^{-5}-2z^{3}a^{-7}-5a^{2}z^{2}-9z^{2}a^{-2}+z^{2}a^{-4}+2z^{2}a^{-6}-13z^{2}-a^{3}z-4az-6za^{-1}-4za^{-3}-za^{-5}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a282,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11a282,}

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["K11a81"];

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]= { $-t^{4}+6t^{3}-16t^{2}+26t-29+26t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$ , $q^{7}-4q^{6}+8q^{5}-13q^{4}+18q^{3}-20q^{2}+20q-17+13q^{-1}-8q^{-2}+4q^{-3}-q^{-4}$ }

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]= {K11a282,}

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

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

Out[6]= {K11a282,}

Vassiliev invariants

V₂ and V₃:

(0, 0)

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
$0$	$0$	$0$	$0$	$0$	$0$	$32$	$32$	$0$	$0$	$0$	$0$	$0$	$32$	$176$	$-160$	$16$	$-48$

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

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j-2r=s+1

or

j-2r=s-1

, where

s=

2 is the signature of K11a81. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

15

1

13

3

-3

11

5

1

4

9

8

3

-5

7

10

5

10

8

-2

3

10

0

1

8

11

3

-1

5

9

-4

-3

3

8

5

-5

1

5

-4

-7

3

-9

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$

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.

K11a80

K11a82

@@ Line 16: / Line 16: @@
 k = 81 |
 KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-6,2,-1,3,-10,4,-9,5,-2,6,-3,7,-4,8,-11,9,-7,10,-8,11,-5/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:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]]</td></tr>
@@ Line 58: / Line 58: @@
          <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, Alternating, 81]]</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>

K11a81: Difference between revisions

Latest revision as of 02:41, 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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