K11a162: Difference between revisions

Latest revision as of 01:54, 3 September 2005

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a162 at Knotilus!
	[edit K11a162 Quick Notes]

[edit K11a162 Further Notes and Views]

Knot presentations

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

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:K11a162/ThurstonBennequinNumber
Hyperbolic Volume	18.245
A-Polynomial	See Data:K11a162/A-polynomial

[edit Notes for K11a162'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 K11a162's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+7t^{3}-20t^{2}+35t-41+35t^{-1}-20t^{-2}+7t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-z^{6}+2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 167, -2 }
Jones polynomial	$q^{3}-5q^{2}+11q-17+24q^{-1}-27q^{-2}+27q^{-3}-23q^{-4}+17q^{-5}-10q^{-6}+4q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+2a^{4}z^{6}-4a^{2}z^{6}+z^{6}-a^{6}z^{4}+6a^{4}z^{4}-5a^{2}z^{4}+2z^{4}-2a^{6}z^{2}+5a^{4}z^{2}-a^{2}z^{2}-a^{6}+a^{4}+a^{2}$
Kauffman polynomial (db, data sources)	$3a^{4}z^{10}+3a^{2}z^{10}+9a^{5}z^{9}+18a^{3}z^{9}+9az^{9}+12a^{6}z^{8}+20a^{4}z^{8}+18a^{2}z^{8}+10z^{8}+9a^{7}z^{7}-2a^{5}z^{7}-27a^{3}z^{7}-11az^{7}+5z^{7}a^{-1}+4a^{8}z^{6}-18a^{6}z^{6}-53a^{4}z^{6}-54a^{2}z^{6}+z^{6}a^{-2}-22z^{6}+a^{9}z^{5}-13a^{7}z^{5}-18a^{5}z^{5}-2a^{3}z^{5}-7az^{5}-9z^{5}a^{-1}-4a^{8}z^{4}+13a^{6}z^{4}+42a^{4}z^{4}+39a^{2}z^{4}-z^{4}a^{-2}+13z^{4}-a^{9}z^{3}+9a^{7}z^{3}+18a^{5}z^{3}+13a^{3}z^{3}+8az^{3}+3z^{3}a^{-1}+a^{8}z^{2}-6a^{6}z^{2}-13a^{4}z^{2}-7a^{2}z^{2}-z^{2}-3a^{7}z-5a^{5}z-3a^{3}z-az+a^{6}+a^{4}-a^{2}$
The A2 invariant	$-q^{24}+q^{22}-3q^{18}+4q^{16}-4q^{14}+2q^{12}+2q^{10}-3q^{8}+6q^{6}-5q^{4}+5q^{2}-2q^{-2}+3q^{-4}-3q^{-6}+q^{-8}$
The G2 invariant	$q^{128}-3q^{126}+7q^{124}-13q^{122}+17q^{120}-19q^{118}+12q^{116}+11q^{114}-46q^{112}+94q^{110}-138q^{108}+151q^{106}-118q^{104}+17q^{102}+152q^{100}-344q^{98}+509q^{96}-563q^{94}+427q^{92}-103q^{90}-372q^{88}+858q^{86}-1159q^{84}+1122q^{82}-679q^{80}-70q^{78}+863q^{76}-1399q^{74}+1437q^{72}-936q^{70}+69q^{68}+791q^{66}-1265q^{64}+1134q^{62}-429q^{60}-505q^{58}+1243q^{56}-1419q^{54}+900q^{52}+100q^{50}-1198q^{48}+1938q^{46}-1983q^{44}+1306q^{42}-122q^{40}-1126q^{38}+1989q^{36}-2169q^{34}+1607q^{32}-550q^{30}-614q^{28}+1451q^{26}-1645q^{24}+1181q^{22}-254q^{20}-695q^{18}+1248q^{16}-1177q^{14}+500q^{12}+447q^{10}-1235q^{8}+1534q^{6}-1190q^{4}+367q^{2}+595-1310q^{-2}+1523q^{-4}-1208q^{-6}+535q^{-8}+190q^{-10}-721q^{-12}+919q^{-14}-794q^{-16}+480q^{-18}-114q^{-20}-162q^{-22}+286q^{-24}-290q^{-26}+206q^{-28}-102q^{-30}+23q^{-32}+28q^{-34}-42q^{-36}+36q^{-38}-24q^{-40}+11q^{-42}-4q^{-44}+q^{-46}$

Further Quantum Invariants

K11a162 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["K11a162"];

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

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

Out[4]= $-t^{4}+7t^{3}-20t^{2}+35t-41+35t^{-1}-20t^{-2}+7t^{-3}-t^{-4}$

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

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

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]= { 167, -2 }

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

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

Out[8]= $q^{3}-5q^{2}+11q-17+24q^{-1}-27q^{-2}+27q^{-3}-23q^{-4}+17q^{-5}-10q^{-6}+4q^{-7}-q^{-8}$

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

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

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

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

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

Out[10]= $3a^{4}z^{10}+3a^{2}z^{10}+9a^{5}z^{9}+18a^{3}z^{9}+9az^{9}+12a^{6}z^{8}+20a^{4}z^{8}+18a^{2}z^{8}+10z^{8}+9a^{7}z^{7}-2a^{5}z^{7}-27a^{3}z^{7}-11az^{7}+5z^{7}a^{-1}+4a^{8}z^{6}-18a^{6}z^{6}-53a^{4}z^{6}-54a^{2}z^{6}+z^{6}a^{-2}-22z^{6}+a^{9}z^{5}-13a^{7}z^{5}-18a^{5}z^{5}-2a^{3}z^{5}-7az^{5}-9z^{5}a^{-1}-4a^{8}z^{4}+13a^{6}z^{4}+42a^{4}z^{4}+39a^{2}z^{4}-z^{4}a^{-2}+13z^{4}-a^{9}z^{3}+9a^{7}z^{3}+18a^{5}z^{3}+13a^{3}z^{3}+8az^{3}+3z^{3}a^{-1}+a^{8}z^{2}-6a^{6}z^{2}-13a^{4}z^{2}-7a^{2}z^{2}-z^{2}-3a^{7}z-5a^{5}z-3a^{3}z-az+a^{6}+a^{4}-a^{2}$

"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["K11a162"];

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]= { $-t^{4}+7t^{3}-20t^{2}+35t-41+35t^{-1}-20t^{-2}+7t^{-3}-t^{-4}$ , $q^{3}-5q^{2}+11q-17+24q^{-1}-27q^{-2}+27q^{-3}-23q^{-4}+17q^{-5}-10q^{-6}+4q^{-7}-q^{-8}$ }

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:

(2, -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

8

-24

32

{\frac {220}{3}}

-{\frac {4}{3}}

-192

-304

0

-56

{\frac {256}{3}}

288

{\frac {1760}{3}}

-{\frac {32}{3}}

{\frac {18991}{15}}

-{\frac {724}{15}}

{\frac {16324}{45}}

{\frac {833}{9}}

{\frac {271}{15}}

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 K11a162. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

4

-4

3

7

1

6

1

10

4

-6

-1

14

7

-3

14

11

-3

-5

13

0

-7

10

14

4

-9

7

13

-6

-11

3

10

7

-13

1

7

-6

-15

3

-17

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-3$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=-1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$

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.

K11a161

K11a163

@@ Line 16: / Line 16: @@
 k = 162 |
 KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-6,2,-1,3,-7,4,-11,5,-2,6,-10,7,-4,8,-5,9,-3,10,-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: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]]</td></tr>
 <tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.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, 162]]</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>

K11a162: Difference between revisions

Latest revision as of 01:54, 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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