K11a19

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K11a18

K11a20

Contents

Image:K11a19.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a19's page at Knotilus!

Visit K11a19's page at the original Knot Atlas!

[edit K11a19 Quick Notes]


[edit K11a19 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X8493 X12,5,13,6 X2837 X16,9,17,10 X18,12,19,11 X6,13,7,14 X20,16,21,15 X22,17,1,18 X14,20,15,19 X10,22,11,21
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, 5, -11, 6, -3, 7, -10, 8, -5, 9, -6, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 8 12 2 16 18 6 20 22 14 10
A Braid Representative
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gif
A Morse Link Presentation Image:K11a19_ML.gif

[edit] Three dimensional invariants

Symmetry type Chiral
Unknotting number {1,2}
3-genus 4
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a19/ThurstonBennequinNumber
Hyperbolic Volume 17.7863
A-Polynomial See Data:K11a19/A-polynomial

[edit Notes for K11a19's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant -2

[edit Notes for K11a19's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t4 + 6t3−18t2 + 33t−39 + 33t−1−18t−2 + 6t−3−t−4
Conway polynomial −z8−2z6−2z4−z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 155, 2 }
Jones polynomial −q8 + 4q7−9q6 + 16q5−22q4 + 25q3−25q2 + 22q−16 + 10q−1−4q−2 + q−3
HOMFLY-PT polynomial (db, data sources) −z8a−2−5z6a−2 + 2z6a−4 + z6−11z4a−2 + 7z4a−4−z4a−6 + 3z4−12z2a−2 + 9z2a−4−2z2a−6 + 4z2−5a−2 + 4a−4−a−6 + 3
Kauffman polynomial (db, data sources) 2z10a−2 + 2z10a−4 + 7z9a−1 + 14z9a−3 + 7z9a−5 + 19z8a−2 + 21z8a−4 + 10z8a−6 + 8z8 + 4az7−6z7a−1−14z7a−3 + 4z7a−5 + 8z7a−7 + a2z6−55z6a−2−52z6a−4−12z6a−6 + 4z6a−8−18z6−8az5−11z5a−1−17z5a−3−26z5a−5−11z5a−7 + z5a−9−2a2z4 + 50z4a−2 + 43z4a−4 + 5z4a−6−5z4a−8 + 15z4 + 5az3 + 11z3a−1 + 21z3a−3 + 23z3a−5 + 7z3a−7−z3a−9 + a2z2−24z2a−2−18z2a−4−z2a−6 + 2z2a−8−8z2−az−3za−1−6za−3−6za−5−2za−7 + 5a−2 + 4a−4 + a−6 + 3
The A2 invariant q8−2q6 + 4q4−q2 + 4q−2−6q−4 + 4q−6−4q−8 + q−10 + 2q−12−3q−14 + 5q−16−2q−18 + q−22−q−24
The G2 invariant q46−3q44 + 8q42−16q40 + 23q38−28q36 + 20q34 + 11q32−66q30 + 145q28−216q26 + 228q24−143q22−63q20 + 357q18−625q16 + 753q14−615q12 + 190q10 + 409q8−975q6 + 1270q4−1120q2 + 547 + 253q−2−963q−4 + 1286q−6−1080q−8 + 449q−10 + 337q−12−920q−14 + 1032q−16−632q−18−121q−20 + 884q−22−1316q−24 + 1205q−26−568q−28−380q−30 + 1275q−32−1781q−34 + 1685q−36−1007q−38−25q−40 + 1032q−42−1653q−44 + 1665q−46−1082q−48 + 174q−50 + 693q−52−1162q−54 + 1067q−56−488q−58−276q−60 + 875q−62−1027q−64 + 679q−66 + 6q−68−720q−70 + 1166q−72−1164q−74 + 750q−76−106q−78−527q−80 + 917q−82−982q−84 + 755q−86−353q−88−53q−90 + 345q−92−475q−94 + 442q−96−312q−98 + 147q−100−94q−104 + 128q−106−121q−108 + 88q−110−46q−112 + 15q−114 + 8q−116−17q−118 + 16q−120−13q−122 + 7q−124−3q−126 + q−128
Further Quantum Invariants
K11a19 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.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["K11a19"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t4 + 6t3−18t2 + 33t−39 + 33t−1−18t−2 + 6t−3−t−4
In[5]:= Conway[K][z]
Out[5]= −z8−2z6−2z4−z2 + 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]= { 155, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q8 + 4q7−9q6 + 16q5−22q4 + 25q3−25q2 + 22q−16 + 10q−1−4q−2 + q−3
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z8a−2−5z6a−2 + 2z6a−4 + z6−11z4a−2 + 7z4a−4−z4a−6 + 3z4−12z2a−2 + 9z2a−4−2z2a−6 + 4z2−5a−2 + 4a−4−a−6 + 3
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 2z10a−2 + 2z10a−4 + 7z9a−1 + 14z9a−3 + 7z9a−5 + 19z8a−2 + 21z8a−4 + 10z8a−6 + 8z8 + 4az7−6z7a−1−14z7a−3 + 4z7a−5 + 8z7a−7 + a2z6−55z6a−2−52z6a−4−12z6a−6 + 4z6a−8−18z6−8az5−11z5a−1−17z5a−3−26z5a−5−11z5a−7 + z5a−9−2a2z4 + 50z4a−2 + 43z4a−4 + 5z4a−6−5z4a−8 + 15z4 + 5az3 + 11z3a−1 + 21z3a−3 + 23z3a−5 + 7z3a−7−z3a−9 + a2z2−24z2a−2−18z2a−4−z2a−6 + 2z2a−8−8z2−az−3za−1−6za−3−6za−5−2za−7 + 5a−2 + 4a−4 + a−6 + 3

[edit] "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a25, K11a281,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a25,}

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.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["K11a19"];
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]= { −t4 + 6t3−18t2 + 33t−39 + 33t−1−18t−2 + 6t−3−t−4, −q8 + 4q7−9q6 + 16q5−22q4 + 25q3−25q2 + 22q−16 + 10q−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]= {K11a25, K11a281,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11a25,}

[edit] Vassiliev invariants

V2 and V3: (-1, 0)

[edit] Khovanov Homology

The coefficients of the monomials trqj are shown, along with their alternating sums χ (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 K11a19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-101234567χ
17           1-1
15          3 3
13         61 -5
11        103  7
9       126   -6
7      1310    3
5     1212     0
3    1013      -3
1   713       6
-1  39        -6
-3 17         6
-5 3          -3
-71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 1 i = 3
r = −4 {\mathbb Z}
r = −3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
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}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r = 3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r = 4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r = 5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 6 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 7 {\mathbb Z}_2 {\mathbb Z}

[edit] Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.


[edit] 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).

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K11a18

K11a20

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