K11a15: Difference between revisions

Latest revision as of 01:50, 3 September 2005

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a15 at Knotilus!
	[edit K11a15 Quick Notes]

[edit K11a15 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_12,5,13,6 X₂₈₃₇ X_14,9,15,10 X_18,12,19,11 X_6,13,7,14 X_20,15,21,16 X_22,17,1,18 X_10,20,11,19 X_16,21,17,22
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -3, 7, -5, 8, -11, 9, -6, 10, -8, 11, -9
Dowker-Thistlethwaite code	4 8 12 2 14 18 6 20 22 10 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:K11a15/ThurstonBennequinNumber
Hyperbolic Volume	15.1288
A-Polynomial	See Data:K11a15/A-polynomial

[edit Notes for K11a15'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 K11a15's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-13t^{2}+22t-25+22t^{-1}-13t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-3z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 107, -2 }
Jones polynomial	$-q^{4}+3q^{3}-6q^{2}+11q-14+17q^{-1}-17q^{-2}+15q^{-3}-12q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-15a^{2}z^{4}-z^{4}a^{-2}+9z^{4}+6a^{4}z^{2}-18a^{2}z^{2}-3z^{2}a^{-2}+14z^{2}+3a^{4}-8a^{2}-2a^{-2}+8$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+5a^{3}z^{9}+8az^{9}+3z^{9}a^{-1}+9a^{4}z^{8}+15a^{2}z^{8}+3z^{8}a^{-2}+9z^{8}+9a^{5}z^{7}+a^{3}z^{7}-15az^{7}-6z^{7}a^{-1}+z^{7}a^{-3}+6a^{6}z^{6}-16a^{4}z^{6}-52a^{2}z^{6}-12z^{6}a^{-2}-42z^{6}+3a^{7}z^{5}-15a^{5}z^{5}-25a^{3}z^{5}-9az^{5}-6z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-6a^{6}z^{4}+13a^{4}z^{4}+58a^{2}z^{4}+16z^{4}a^{-2}+54z^{4}-2a^{7}z^{3}+14a^{5}z^{3}+29a^{3}z^{3}+23az^{3}+15z^{3}a^{-1}+5z^{3}a^{-3}-a^{8}z^{2}+3a^{6}z^{2}-7a^{4}z^{2}-33a^{2}z^{2}-9z^{2}a^{-2}-31z^{2}-5a^{5}z-11a^{3}z-10az-6za^{-1}-2za^{-3}+3a^{4}+8a^{2}+2a^{-2}+8$
The A2 invariant	$q^{20}-q^{18}+3q^{16}-q^{14}-q^{12}+q^{10}-5q^{8}+2q^{6}-3q^{4}+2q^{2}+3+4q^{-4}-q^{-6}-q^{-12}$
The G2 invariant	$q^{114}-2q^{112}+4q^{110}-6q^{108}+6q^{106}-5q^{104}+9q^{100}-19q^{98}+29q^{96}-37q^{94}+33q^{92}-20q^{90}-5q^{88}+44q^{86}-78q^{84}+105q^{82}-114q^{80}+87q^{78}-31q^{76}-56q^{74}+155q^{72}-222q^{70}+240q^{68}-180q^{66}+52q^{64}+114q^{62}-253q^{60}+318q^{58}-267q^{56}+116q^{54}+78q^{52}-231q^{50}+278q^{48}-189q^{46}+10q^{44}+180q^{42}-295q^{40}+259q^{38}-97q^{36}-144q^{34}+352q^{32}-446q^{30}+366q^{28}-152q^{26}-136q^{24}+378q^{22}-501q^{20}+449q^{18}-254q^{16}-18q^{14}+262q^{12}-389q^{10}+367q^{8}-197q^{6}-28q^{4}+220q^{2}-291+223q^{-2}-32q^{-4}-173q^{-6}+318q^{-8}-320q^{-10}+188q^{-12}+29q^{-14}-235q^{-16}+359q^{-18}-346q^{-20}+222q^{-22}-39q^{-24}-136q^{-26}+236q^{-28}-246q^{-30}+181q^{-32}-80q^{-34}-17q^{-36}+76q^{-38}-97q^{-40}+81q^{-42}-48q^{-44}+17q^{-46}+5q^{-48}-16q^{-50}+14q^{-52}-11q^{-54}+6q^{-56}-2q^{-58}+q^{-60}$

Further Quantum Invariants

K11a15 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["K11a15"];

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

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

Out[4]= $-t^{4}+5t^{3}-13t^{2}+22t-25+22t^{-1}-13t^{-2}+5t^{-3}-t^{-4}$

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

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

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]= { 107, -2 }

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

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

Out[8]= $-q^{4}+3q^{3}-6q^{2}+11q-14+17q^{-1}-17q^{-2}+15q^{-3}-12q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$

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

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

Out[9]= $-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-15a^{2}z^{4}-z^{4}a^{-2}+9z^{4}+6a^{4}z^{2}-18a^{2}z^{2}-3z^{2}a^{-2}+14z^{2}+3a^{4}-8a^{2}-2a^{-2}+8$

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

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

Out[10]= $a^{2}z^{10}+z^{10}+5a^{3}z^{9}+8az^{9}+3z^{9}a^{-1}+9a^{4}z^{8}+15a^{2}z^{8}+3z^{8}a^{-2}+9z^{8}+9a^{5}z^{7}+a^{3}z^{7}-15az^{7}-6z^{7}a^{-1}+z^{7}a^{-3}+6a^{6}z^{6}-16a^{4}z^{6}-52a^{2}z^{6}-12z^{6}a^{-2}-42z^{6}+3a^{7}z^{5}-15a^{5}z^{5}-25a^{3}z^{5}-9az^{5}-6z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-6a^{6}z^{4}+13a^{4}z^{4}+58a^{2}z^{4}+16z^{4}a^{-2}+54z^{4}-2a^{7}z^{3}+14a^{5}z^{3}+29a^{3}z^{3}+23az^{3}+15z^{3}a^{-1}+5z^{3}a^{-3}-a^{8}z^{2}+3a^{6}z^{2}-7a^{4}z^{2}-33a^{2}z^{2}-9z^{2}a^{-2}-31z^{2}-5a^{5}z-11a^{3}z-10az-6za^{-1}-2za^{-3}+3a^{4}+8a^{2}+2a^{-2}+8$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

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["K11a15"];

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]= { $-t^{4}+5t^{3}-13t^{2}+22t-25+22t^{-1}-13t^{-2}+5t^{-3}-t^{-4}$ , $-q^{4}+3q^{3}-6q^{2}+11q-14+17q^{-1}-17q^{-2}+15q^{-3}-12q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$ }

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:

(-1, 3)

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

24

8

{\frac {34}{3}}

{\frac {86}{3}}

-96

-272

-96

-104

-{\frac {32}{3}}

288

-{\frac {136}{3}}

-{\frac {344}{3}}

{\frac {22289}{30}}

{\frac {3622}{15}}

{\frac {778}{45}}

{\frac {2383}{18}}

-{\frac {2671}{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=

-2 is the signature of K11a15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

9

1

-1

7

2

5

4

1

-3

3

7

2

5

1

7

4

-3

-1

10

7

3

-3

8

0

-5

7

9

-2

-7

5

8

3

-9

2

7

-5

-11

1

5

4

-13

2

-2

-15

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$

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.

K11a14

K11a16

@@ Line 16: / Line 16: @@
 k = 15 |
 KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,2,-1,3,-7,4,-2,5,-10,6,-3,7,-5,8,-11,9,-6,10,-8,11,-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:BraidPart0.gif]][[Image:BraidPart1.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:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.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, 15]]</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>

K11a15: Difference between revisions

Latest revision as of 01:50, 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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