K11a71: 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 K11a71 at Knotilus!
	[edit K11a71 Quick Notes]

[edit K11a71 Further Notes and Views]

Knot presentations

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

A Braid Representative

A Morse Link Presentation

Three dimensional invariants

Symmetry type	Chiral
Unknotting number	1
3-genus	4
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a71/ThurstonBennequinNumber
Hyperbolic Volume	17.3873
A-Polynomial	See Data:K11a71/A-polynomial

[edit Notes for K11a71'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 K11a71's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-18t^{2}+34t-41+34t^{-1}-18t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}-2z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 159, 2 }
Jones polynomial	$-q^{8}+5q^{7}-11q^{6}+17q^{5}-23q^{4}+26q^{3}-25q^{2}+22q-15+9q^{-1}-4q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-5z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-10z^{4}a^{-2}+6z^{4}a^{-4}-z^{4}a^{-6}+3z^{4}-7z^{2}a^{-2}+5z^{2}a^{-4}-z^{2}a^{-6}+3z^{2}+1$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}a^{-4}+6z^{9}a^{-1}+14z^{9}a^{-3}+8z^{9}a^{-5}+17z^{8}a^{-2}+23z^{8}a^{-4}+13z^{8}a^{-6}+7z^{8}+4az^{7}-3z^{7}a^{-1}-13z^{7}a^{-3}+5z^{7}a^{-5}+11z^{7}a^{-7}+a^{2}z^{6}-47z^{6}a^{-2}-53z^{6}a^{-4}-17z^{6}a^{-6}+5z^{6}a^{-8}-15z^{6}-9az^{5}-14z^{5}a^{-1}-18z^{5}a^{-3}-29z^{5}a^{-5}-15z^{5}a^{-7}+z^{5}a^{-9}-2a^{2}z^{4}+39z^{4}a^{-2}+35z^{4}a^{-4}+5z^{4}a^{-6}-4z^{4}a^{-8}+11z^{4}+6az^{3}+14z^{3}a^{-1}+20z^{3}a^{-3}+17z^{3}a^{-5}+5z^{3}a^{-7}+a^{2}z^{2}-12z^{2}a^{-2}-8z^{2}a^{-4}-z^{2}a^{-6}-4z^{2}-az-3za^{-1}-3za^{-3}-za^{-5}+1$
The A2 invariant	$q^{8}-2q^{6}+3q^{4}-2q^{2}-1+5q^{-2}-4q^{-4}+6q^{-6}-2q^{-8}+q^{-12}-5q^{-14}+4q^{-16}-2q^{-18}+2q^{-22}-q^{-24}$
The G2 invariant	$q^{46}-3q^{44}+8q^{42}-16q^{40}+22q^{38}-25q^{36}+14q^{34}+18q^{32}-65q^{30}+127q^{28}-176q^{26}+178q^{24}-110q^{22}-51q^{20}+277q^{18}-495q^{16}+619q^{14}-551q^{12}+254q^{10}+221q^{8}-733q^{6}+1094q^{4}-1121q^{2}+759-102q^{-2}-637q^{-4}+1158q^{-6}-1252q^{-8}+878q^{-10}-176q^{-12}-542q^{-14}+971q^{-16}-920q^{-18}+406q^{-20}+341q^{-22}-974q^{-24}+1199q^{-26}-873q^{-28}+94q^{-30}+840q^{-32}-1553q^{-34}+1765q^{-36}-1353q^{-38}+450q^{-40}+627q^{-42}-1496q^{-44}+1839q^{-46}-1553q^{-48}+761q^{-50}+212q^{-52}-989q^{-54}+1279q^{-56}-1014q^{-58}+349q^{-60}+402q^{-62}-898q^{-64}+913q^{-66}-467q^{-68}-244q^{-70}+900q^{-72}-1206q^{-74}+1053q^{-76}-495q^{-78}-230q^{-80}+847q^{-82}-1157q^{-84}+1079q^{-86}-685q^{-88}+156q^{-90}+320q^{-92}-610q^{-94}+664q^{-96}-522q^{-98}+289q^{-100}-49q^{-102}-126q^{-104}+202q^{-106}-205q^{-108}+147q^{-110}-76q^{-112}+22q^{-114}+16q^{-116}-28q^{-118}+27q^{-120}-20q^{-122}+10q^{-124}-4q^{-126}+q^{-128}$

Further Quantum Invariants

K11a71 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["K11a71"];

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

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

Out[4]= $-t^{4}+6t^{3}-18t^{2}+34t-41+34t^{-1}-18t^{-2}+6t^{-3}-t^{-4}$

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

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

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]= { 159, 2 }

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

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

Out[8]= $-q^{8}+5q^{7}-11q^{6}+17q^{5}-23q^{4}+26q^{3}-25q^{2}+22q-15+9q^{-1}-4q^{-2}+q^{-3}$

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

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

Out[9]= $-z^{8}a^{-2}-5z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-10z^{4}a^{-2}+6z^{4}a^{-4}-z^{4}a^{-6}+3z^{4}-7z^{2}a^{-2}+5z^{2}a^{-4}-z^{2}a^{-6}+3z^{2}+1$

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

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

Out[10]= $2z^{10}a^{-2}+2z^{10}a^{-4}+6z^{9}a^{-1}+14z^{9}a^{-3}+8z^{9}a^{-5}+17z^{8}a^{-2}+23z^{8}a^{-4}+13z^{8}a^{-6}+7z^{8}+4az^{7}-3z^{7}a^{-1}-13z^{7}a^{-3}+5z^{7}a^{-5}+11z^{7}a^{-7}+a^{2}z^{6}-47z^{6}a^{-2}-53z^{6}a^{-4}-17z^{6}a^{-6}+5z^{6}a^{-8}-15z^{6}-9az^{5}-14z^{5}a^{-1}-18z^{5}a^{-3}-29z^{5}a^{-5}-15z^{5}a^{-7}+z^{5}a^{-9}-2a^{2}z^{4}+39z^{4}a^{-2}+35z^{4}a^{-4}+5z^{4}a^{-6}-4z^{4}a^{-8}+11z^{4}+6az^{3}+14z^{3}a^{-1}+20z^{3}a^{-3}+17z^{3}a^{-5}+5z^{3}a^{-7}+a^{2}z^{2}-12z^{2}a^{-2}-8z^{2}a^{-4}-z^{2}a^{-6}-4z^{2}-az-3za^{-1}-3za^{-3}-za^{-5}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a248,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11a248,}

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["K11a71"];

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]= { $-t^{4}+6t^{3}-18t^{2}+34t-41+34t^{-1}-18t^{-2}+6t^{-3}-t^{-4}$ , $-q^{8}+5q^{7}-11q^{6}+17q^{5}-23q^{4}+26q^{3}-25q^{2}+22q-15+9q^{-1}-4q^{-2}+q^{-3}$ }

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]= {K11a248,}

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

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

Out[6]= {K11a248,}

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
$0$	$0$	$0$	$16$	$16$	$0$	$32$	$0$	$32$	$0$	$0$	$0$	$0$	$40$	${\frac {176}{3}}$	$-{\frac {160}{3}}$	${\frac {104}{3}}$	$-40$

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 K11a71. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

17

1

-1

15

4

13

7

1

-6

11

10

4

6

9

13

7

-6

7

13

10

3

5

12

13

1

3

10

13

-3

1

6

13

7

-1

3

9

-6

-3

1

6

5

-5

3

-3

-7

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$

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.

K11a70

K11a72

@@ Line 16: / Line 16: @@
 k = 71 |
 KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-5,2,-1,3,-9,4,-10,5,-2,6,-3,7,-4,8,-11,9,-7,10,-8,11,-6/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:BraidPart1.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:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.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, 71]]</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>

K11a71: 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

Navigation menu

Page actions

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