K11a33: Difference between revisions

Latest revision as of 01:42, 3 September 2005

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a33 at Knotilus!
	[edit K11a33 Quick Notes]

[edit K11a33 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_14,5,15,6 X₂₈₃₇ X_16,9,17,10 X_18,11,19,12 X_20,14,21,13 X_6,15,7,16 X_10,17,11,18 X_22,19,1,20 X_12,22,13,21
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, 5, -9, 6, -11, 7, -3, 8, -5, 9, -6, 10, -7, 11, -10
Dowker-Thistlethwaite code	4 8 14 2 16 18 20 6 10 22 12

A Braid Representative

A Morse Link Presentation

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	$\{1,2\}$
3-genus	4
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a33/ThurstonBennequinNumber
Hyperbolic Volume	13.8946
A-Polynomial	See Data:K11a33/A-polynomial

[edit Notes for K11a33'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 K11a33's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-12t^{2}+19t-21+19t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-2z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 95, -2 }
Jones polynomial	$-q^{4}+3q^{3}-6q^{2}+10q-12+15q^{-1}-15q^{-2}+13q^{-3}-10q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-14a^{2}z^{4}-z^{4}a^{-2}+9z^{4}+5a^{4}z^{2}-15a^{2}z^{2}-3z^{2}a^{-2}+13z^{2}+2a^{4}-6a^{2}-2a^{-2}+7$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+4a^{3}z^{9}+7az^{9}+3z^{9}a^{-1}+6a^{4}z^{8}+10a^{2}z^{8}+3z^{8}a^{-2}+7z^{8}+6a^{5}z^{7}-2a^{3}z^{7}-16az^{7}-7z^{7}a^{-1}+z^{7}a^{-3}+5a^{6}z^{6}-8a^{4}z^{6}-37a^{2}z^{6}-12z^{6}a^{-2}-36z^{6}+3a^{7}z^{5}-6a^{5}z^{5}-9a^{3}z^{5}-4z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-5a^{6}z^{4}+5a^{4}z^{4}+42a^{2}z^{4}+15z^{4}a^{-2}+46z^{4}-3a^{7}z^{3}+2a^{5}z^{3}+10a^{3}z^{3}+13az^{3}+13z^{3}a^{-1}+5z^{3}a^{-3}-a^{8}z^{2}+2a^{6}z^{2}-4a^{4}z^{2}-24a^{2}z^{2}-8z^{2}a^{-2}-25z^{2}+a^{7}z-4a^{3}z-6az-5za^{-1}-2za^{-3}+2a^{4}+6a^{2}+2a^{-2}+7$
The A2 invariant	$q^{20}-q^{18}+2q^{16}-q^{14}-q^{12}+q^{10}-4q^{8}+2q^{6}-2q^{4}+2q^{2}+3+3q^{-4}-q^{-6}-q^{-12}$
The G2 invariant	$q^{114}-2q^{112}+4q^{110}-6q^{108}+5q^{106}-4q^{104}-2q^{102}+11q^{100}-20q^{98}+28q^{96}-30q^{94}+21q^{92}-4q^{90}-18q^{88}+46q^{86}-64q^{84}+71q^{82}-62q^{80}+34q^{78}+5q^{76}-50q^{74}+94q^{72}-120q^{70}+122q^{68}-91q^{66}+28q^{64}+51q^{62}-120q^{60}+164q^{58}-153q^{56}+91q^{54}+2q^{52}-95q^{50}+142q^{48}-123q^{46}+48q^{44}+53q^{42}-131q^{40}+140q^{38}-78q^{36}-43q^{34}+164q^{32}-242q^{30}+220q^{28}-117q^{26}-45q^{24}+200q^{22}-292q^{20}+287q^{18}-190q^{16}+33q^{14}+123q^{12}-226q^{10}+246q^{8}-169q^{6}+45q^{4}+87q^{2}-158+157q^{-2}-71q^{-4}-46q^{-6}+152q^{-8}-189q^{-10}+143q^{-12}-26q^{-14}-108q^{-16}+214q^{-18}-235q^{-20}+177q^{-22}-63q^{-24}-68q^{-26}+158q^{-28}-189q^{-30}+156q^{-32}-82q^{-34}+q^{-36}+56q^{-38}-82q^{-40}+74q^{-42}-48q^{-44}+19q^{-46}+3q^{-48}-15q^{-50}+14q^{-52}-11q^{-54}+6q^{-56}-2q^{-58}+q^{-60}$

Further Quantum Invariants

K11a33 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["K11a33"];

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

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

Out[4]= $-t^{4}+5t^{3}-12t^{2}+19t-21+19t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$

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

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

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]= { 95, -2 }

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

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

Out[8]= $-q^{4}+3q^{3}-6q^{2}+10q-12+15q^{-1}-15q^{-2}+13q^{-3}-10q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$

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

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

Out[9]= $-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-14a^{2}z^{4}-z^{4}a^{-2}+9z^{4}+5a^{4}z^{2}-15a^{2}z^{2}-3z^{2}a^{-2}+13z^{2}+2a^{4}-6a^{2}-2a^{-2}+7$

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

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

Out[10]= $a^{2}z^{10}+z^{10}+4a^{3}z^{9}+7az^{9}+3z^{9}a^{-1}+6a^{4}z^{8}+10a^{2}z^{8}+3z^{8}a^{-2}+7z^{8}+6a^{5}z^{7}-2a^{3}z^{7}-16az^{7}-7z^{7}a^{-1}+z^{7}a^{-3}+5a^{6}z^{6}-8a^{4}z^{6}-37a^{2}z^{6}-12z^{6}a^{-2}-36z^{6}+3a^{7}z^{5}-6a^{5}z^{5}-9a^{3}z^{5}-4z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-5a^{6}z^{4}+5a^{4}z^{4}+42a^{2}z^{4}+15z^{4}a^{-2}+46z^{4}-3a^{7}z^{3}+2a^{5}z^{3}+10a^{3}z^{3}+13az^{3}+13z^{3}a^{-1}+5z^{3}a^{-3}-a^{8}z^{2}+2a^{6}z^{2}-4a^{4}z^{2}-24a^{2}z^{2}-8z^{2}a^{-2}-25z^{2}+a^{7}z-4a^{3}z-6az-5za^{-1}-2za^{-3}+2a^{4}+6a^{2}+2a^{-2}+7$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_116, K11a7, K11a82,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11a82,}

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["K11a33"];

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]= { $-t^{4}+5t^{3}-12t^{2}+19t-21+19t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$ , $-q^{4}+3q^{3}-6q^{2}+10q-12+15q^{-1}-15q^{-2}+13q^{-3}-10q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$ }

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]= {10_116, K11a7, K11a82,}

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

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

Out[6]= {K11a82,}

Vassiliev invariants

V₂ and V₃:

(0, 2)

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

16

0

0

16

0

-{\frac {320}{3}}

-{\frac {128}{3}}

-80

0

128

0

0

448

{\frac {64}{3}}

{\frac {656}{3}}

{\frac {272}{3}}

-16

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 K11a33. 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

6

2

4

1

6

4

-2

-1

9

6

3

-3

7

0

-5

6

8

-2

-7

4

7

3

-9

2

6

-4

-11

1

4

3

-13

2

-2

-15

1

Integral Khovanov Homology

(db, data source)

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

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.

K11a32

K11a34

@@ Line 16: / Line 16: @@
 k = 33 |
 KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,2,-1,3,-8,4,-2,5,-9,6,-11,7,-3,8,-5,9,-6,10,-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: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:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.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, 33]]</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>

K11a33: Difference between revisions

Latest revision as of 01:42, 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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