K11a99

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K11a98

K11a100

Contents

Image:K11a99.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a99's page at Knotilus!

Visit K11a99's page at the original Knot Atlas!

[edit K11a99 Quick Notes]


[edit K11a99 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X20,8,21,7 X16,10,17,9 X2,11,3,12 X18,13,19,14 X8,16,9,15 X22,17,1,18 X6,20,7,19 X14,21,15,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -11, 8, -5, 9, -7, 10, -4, 11, -9
Dowker-Thistlethwaite code 4 10 12 20 16 2 18 8 22 6 14
A Braid Representative
Image:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gif
A Morse Link Presentation Image:K11a99_ML.gif

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number {1,2}
3-genus 4
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a99/ThurstonBennequinNumber
Hyperbolic Volume 16.6086
A-Polynomial See Data:K11a99/A-polynomial

[edit Notes for K11a99'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 K11a99's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t4 + 6t3−17t2 + 28t−31 + 28t−1−17t−2 + 6t−3−t−4
Conway polynomial −z8−2z6−z4−2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 135, 2 }
Jones polynomial q7−4q6 + 9q5−14q4 + 19q3−22q2 + 21q−18 + 14q−1−8q−2 + 4q−3−q−4
HOMFLY-PT polynomial (db, data sources) −z8a−2−5z6a−2 + z6a−4 + 2z6−a2z4−10z4a−2 + 3z4a−4 + 7z4−2a2z2−10z2a−2 + 3z2a−4 + 7z2−4a−2 + 2a−4 + 3
Kauffman polynomial (db, data sources) 2z10a−2 + 2z10 + 5az9 + 13z9a−1 + 8z9a−3 + 4a2z8 + 19z8a−2 + 13z8a−4 + 10z8 + a3z7−12az7−27z7a−1−z7a−3 + 13z7a−5−14a2z6−62z6a−2−18z6a−4 + 9z6a−6−49z6−3a3z5 + az5−25z5a−3−17z5a−5 + 4z5a−7 + 16a2z4 + 55z4a−2 + 7z4a−4−8z4a−6 + z4a−8 + 55z4 + 3a3z3 + 10az3 + 15z3a−1 + 18z3a−3 + 9z3a−5−z3a−7−6a2z2−23z2a−2−5z2a−4 + 3z2a−6−21z2−a3z−3az−3za−1−3za−3−2za−5 + 4a−2 + 2a−4 + 3
The A2 invariant −q12 + q10 + q8−q6 + 4q4−2q2 + 2 + q−2−4q−4 + 3q−6−5q−8 + 3q−10−q−14 + 3q−16−2q−18 + q−20
The G2 invariant q60−3q58 + 9q56−19q54 + 28q52−33q50 + 17q48 + 26q46−91q44 + 165q42−204q40 + 167q38−37q36−174q34 + 394q32−522q30 + 480q28−242q26−135q24 + 516q22−741q20 + 713q18−405q16−48q14 + 467q12−683q10 + 599q8−269q6−156q4 + 498q2−586 + 391q−2 + 18q−4−460q−6 + 748q−8−756q−10 + 454q−12 + 43q−14−573q−16 + 935q−18−987q−20 + 703q−22−172q−24−401q−26 + 798q−28−894q−30 + 643q−32−188q−34−279q−36 + 566q−38−560q−40 + 292q−42 + 105q−44−431q−46 + 544q−48−402q−50 + 68q−52 + 293q−54−544q−56 + 602q−58−444q−60 + 168q−62 + 140q−64−372q−66 + 462q−68−420q−70 + 276q−72−96q−74−65q−76 + 179q−78−227q−80 + 213q−82−152q−84 + 78q−86−5q−88−46q−90 + 68q−92−74q−94 + 57q−96−33q−98 + 13q−100 + 4q−102−10q−104 + 11q−106−10q−108 + 6q−110−3q−112 + q−114
Further Quantum Invariants
K11a99 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["K11a99"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t4 + 6t3−17t2 + 28t−31 + 28t−1−17t−2 + 6t−3−t−4
In[5]:= Conway[K][z]
Out[5]= −z8−2z6−z4−2z2 + 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]= { 135, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q7−4q6 + 9q5−14q4 + 19q3−22q2 + 21q−18 + 14q−1−8q−2 + 4q−3−q−4
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z8a−2−5z6a−2 + z6a−4 + 2z6−a2z4−10z4a−2 + 3z4a−4 + 7z4−2a2z2−10z2a−2 + 3z2a−4 + 7z2−4a−2 + 2a−4 + 3
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 2z10a−2 + 2z10 + 5az9 + 13z9a−1 + 8z9a−3 + 4a2z8 + 19z8a−2 + 13z8a−4 + 10z8 + a3z7−12az7−27z7a−1−z7a−3 + 13z7a−5−14a2z6−62z6a−2−18z6a−4 + 9z6a−6−49z6−3a3z5 + az5−25z5a−3−17z5a−5 + 4z5a−7 + 16a2z4 + 55z4a−2 + 7z4a−4−8z4a−6 + z4a−8 + 55z4 + 3a3z3 + 10az3 + 15z3a−1 + 18z3a−3 + 9z3a−5−z3a−7−6a2z2−23z2a−2−5z2a−4 + 3z2a−6−21z2−a3z−3az−3za−1−3za−3−2za−5 + 4a−2 + 2a−4 + 3

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

Same Alexander/Conway Polynomial: {K11a277,}

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.

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

In[3]:= K = Knot["K11a99"];
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 + 6t3−17t2 + 28t−31 + 28t−1−17t−2 + 6t−3−t−4, q7−4q6 + 9q5−14q4 + 19q3−22q2 + 21q−18 + 14q−1−8q−2 + 4q−3−q−4 }
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]= {K11a277,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {}

[edit] Vassiliev invariants

V2 and V3: (-2, -2)

[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 K11a99. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-10123456χ
15           11
13          3 -3
11         61 5
9        83  -5
7       116   5
5      118    -3
3     1011     -1
1    912      3
-1   59       -4
-3  39        6
-5 15         -4
-7 3          3
-91           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 1 i = 3
r = −5 {\mathbb Z}
r = −4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = −1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r = 0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r = 1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r = 2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r = 3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = 4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 6 {\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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K11a98

K11a100

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