K11a26: Difference between revisions

Latest revision as of 01:52, 3 September 2005

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a26 at Knotilus!
	[edit K11a26 Quick Notes]

[edit K11a26 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_12,5,13,6 X₂₈₃₇ X_20,9,21,10 X_18,12,19,11 X_6,13,7,14 X_10,16,11,15 X_22,17,1,18 X_14,20,15,19 X_16,21,17,22
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, 5, -8, 6, -3, 7, -10, 8, -11, 9, -6, 10, -5, 11, -9
Dowker-Thistlethwaite code	4 8 12 2 20 18 6 10 22 14 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:K11a26/ThurstonBennequinNumber
Hyperbolic Volume	17.9006
A-Polynomial	See Data:K11a26/A-polynomial

[edit Notes for K11a26's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$4$
Rasmussen s-Invariant	0

[edit Notes for K11a26's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+18t^{2}-33t+41-33t^{-1}+18t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}+2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 157, 0 }
Jones polynomial	$q^{6}-4q^{5}+9q^{4}-16q^{3}+22q^{2}-25q+26-22q^{-1}+17q^{-2}-10q^{-3}+4q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+5z^{6}-3a^{2}z^{4}-7z^{4}a^{-2}+z^{4}a^{-4}+11z^{4}-4a^{2}z^{2}-9z^{2}a^{-2}+2z^{2}a^{-4}+12z^{2}-2a^{2}-4a^{-2}+a^{-4}+6$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+8az^{9}+14z^{9}a^{-1}+6z^{9}a^{-3}+12a^{2}z^{8}+18z^{8}a^{-2}+7z^{8}a^{-4}+23z^{8}+9a^{3}z^{7}+2az^{7}-12z^{7}a^{-1}-z^{7}a^{-3}+4z^{7}a^{-5}+4a^{4}z^{6}-19a^{2}z^{6}-50z^{6}a^{-2}-14z^{6}a^{-4}+z^{6}a^{-6}-58z^{6}+a^{5}z^{5}-13a^{3}z^{5}-26az^{5}-24z^{5}a^{-1}-21z^{5}a^{-3}-9z^{5}a^{-5}-4a^{4}z^{4}+15a^{2}z^{4}+42z^{4}a^{-2}+9z^{4}a^{-4}-2z^{4}a^{-6}+50z^{4}-a^{5}z^{3}+9a^{3}z^{3}+25az^{3}+30z^{3}a^{-1}+22z^{3}a^{-3}+7z^{3}a^{-5}+a^{4}z^{2}-8a^{2}z^{2}-18z^{2}a^{-2}-3z^{2}a^{-4}+z^{2}a^{-6}-23z^{2}-3a^{3}z-8az-10za^{-1}-7za^{-3}-2za^{-5}+2a^{2}+4a^{-2}+a^{-4}+6$
The A2 invariant	$-q^{14}+2q^{12}-4q^{10}+2q^{8}+q^{6}-3q^{4}+7q^{2}-3+5q^{-2}-q^{-4}-2q^{-6}+3q^{-8}-5q^{-10}+2q^{-12}-q^{-16}+q^{-18}$
The G2 invariant	$q^{80}-3q^{78}+7q^{76}-13q^{74}+17q^{72}-19q^{70}+12q^{68}+11q^{66}-46q^{64}+94q^{62}-139q^{60}+154q^{58}-123q^{56}+20q^{54}+160q^{52}-369q^{50}+543q^{48}-584q^{46}+408q^{44}-30q^{42}-482q^{40}+949q^{38}-1167q^{36}+1001q^{34}-454q^{32}-314q^{30}+1007q^{28}-1337q^{26}+1155q^{24}-520q^{22}-297q^{20}+930q^{18}-1100q^{16}+723q^{14}+43q^{12}-830q^{10}+1288q^{8}-1170q^{6}+494q^{4}+492q^{2}-1389+1846q^{-2}-1655q^{-4}+877q^{-6}+226q^{-8}-1251q^{-10}+1840q^{-12}-1780q^{-14}+1119q^{-16}-121q^{-18}-823q^{-20}+1338q^{-22}-1252q^{-24}+639q^{-26}+208q^{-28}-899q^{-30}+1117q^{-32}-782q^{-34}+40q^{-36}+763q^{-38}-1275q^{-40}+1280q^{-42}-790q^{-44}+9q^{-46}+733q^{-48}-1173q^{-50}+1190q^{-52}-823q^{-54}+270q^{-56}+253q^{-58}-591q^{-60}+671q^{-62}-539q^{-64}+300q^{-66}-52q^{-68}-120q^{-70}+194q^{-72}-188q^{-74}+134q^{-76}-66q^{-78}+16q^{-80}+15q^{-82}-26q^{-84}+22q^{-86}-16q^{-88}+8q^{-90}-3q^{-92}+q^{-94}$

Further Quantum Invariants

K11a26 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["K11a26"];

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

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

Out[4]= $t^{4}-6t^{3}+18t^{2}-33t+41-33t^{-1}+18t^{-2}-6t^{-3}+t^{-4}$

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

Out[5]= $z^{8}+2z^{6}+2z^{4}+z^{2}+1$

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]= { 157, 0 }

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

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

Out[8]= $q^{6}-4q^{5}+9q^{4}-16q^{3}+22q^{2}-25q+26-22q^{-1}+17q^{-2}-10q^{-3}+4q^{-4}-q^{-5}$

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

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

Out[9]= $z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+5z^{6}-3a^{2}z^{4}-7z^{4}a^{-2}+z^{4}a^{-4}+11z^{4}-4a^{2}z^{2}-9z^{2}a^{-2}+2z^{2}a^{-4}+12z^{2}-2a^{2}-4a^{-2}+a^{-4}+6$

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

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

Out[10]= $2z^{10}a^{-2}+2z^{10}+8az^{9}+14z^{9}a^{-1}+6z^{9}a^{-3}+12a^{2}z^{8}+18z^{8}a^{-2}+7z^{8}a^{-4}+23z^{8}+9a^{3}z^{7}+2az^{7}-12z^{7}a^{-1}-z^{7}a^{-3}+4z^{7}a^{-5}+4a^{4}z^{6}-19a^{2}z^{6}-50z^{6}a^{-2}-14z^{6}a^{-4}+z^{6}a^{-6}-58z^{6}+a^{5}z^{5}-13a^{3}z^{5}-26az^{5}-24z^{5}a^{-1}-21z^{5}a^{-3}-9z^{5}a^{-5}-4a^{4}z^{4}+15a^{2}z^{4}+42z^{4}a^{-2}+9z^{4}a^{-4}-2z^{4}a^{-6}+50z^{4}-a^{5}z^{3}+9a^{3}z^{3}+25az^{3}+30z^{3}a^{-1}+22z^{3}a^{-3}+7z^{3}a^{-5}+a^{4}z^{2}-8a^{2}z^{2}-18z^{2}a^{-2}-3z^{2}a^{-4}+z^{2}a^{-6}-23z^{2}-3a^{3}z-8az-10za^{-1}-7za^{-3}-2za^{-5}+2a^{2}+4a^{-2}+a^{-4}+6$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a24, K11a315,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11a24, K11a315,}

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["K11a26"];

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]= { $t^{4}-6t^{3}+18t^{2}-33t+41-33t^{-1}+18t^{-2}-6t^{-3}+t^{-4}$ , $q^{6}-4q^{5}+9q^{4}-16q^{3}+22q^{2}-25q+26-22q^{-1}+17q^{-2}-10q^{-3}+4q^{-4}-q^{-5}$ }

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]= {K11a24, K11a315,}

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

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

Out[6]= {K11a24, K11a315,}

Vassiliev invariants

V₂ and V₃:

(1, -1)

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

4

-8

8

-{\frac {34}{3}}

-{\frac {38}{3}}

-32

-{\frac {80}{3}}

{\frac {64}{3}}

-8

{\frac {32}{3}}

32

-{\frac {136}{3}}

-{\frac {152}{3}}

{\frac {751}{30}}

-{\frac {902}{15}}

{\frac {3302}{45}}

-{\frac {175}{18}}

{\frac {271}{30}}

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=

0 is the signature of K11a26. 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

13

1

11

3

-3

9

6

1

5

7

10

3

-7

5

12

6

3

13

10

-3

1

13

12

1

-1

10

14

4

-3

7

12

-5

3

10

7

-7

1

7

-6

-9

3

-11

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$

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.

K11a25

K11a27

@@ Line 16: / Line 16: @@
 k = 26 |
 KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,2,-1,3,-7,4,-2,5,-8,6,-3,7,-10,8,-11,9,-6,10,-5,11,-9/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:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr>
 <tr><td>[[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.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, 26]]</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>

K11a26: Difference between revisions

Latest revision as of 01:52, 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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