K11a221

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K11a220

K11a222

Contents

Image:K11a221.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a221's page at Knotilus!

Visit K11a221's page at the original Knot Atlas!

[edit K11a221 Quick Notes]


[edit K11a221 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X18,5,19,6 X20,7,21,8 X14,10,15,9 X16,12,17,11 X2,13,3,14 X10,16,11,15 X22,17,1,18 X6,19,7,20 X8,21,9,22
Gauss code 1, -7, 2, -1, 3, -10, 4, -11, 5, -8, 6, -2, 7, -5, 8, -6, 9, -3, 10, -4, 11, -9
Dowker-Thistlethwaite code 4 12 18 20 14 16 2 10 22 6 8
A Braid Representative
Image:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gif
A Morse Link Presentation Image:K11a221_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:K11a221/ThurstonBennequinNumber
Hyperbolic Volume 12.2428
A-Polynomial See Data:K11a221/A-polynomial

[edit Notes for K11a221'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 K11a221's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t4 + 5t3−10t2 + 15t−17 + 15t−1−10t−2 + 5t−3−t−4
Conway polynomial −z8−3z6 + 4z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 79, -2 }
Jones polynomial q3−3q2 + 5q−8 + 11q−1−12q−2 + 13q−3−10q−4 + 8q−5−5q−6 + 2q−7−q−8
HOMFLY-PT polynomial (db, data sources) −a2z8 + 2a4z6−6a2z6 + z6−a6z4 + 10a4z4−13a2z4 + 4z4−4a6z2 + 16a4z2−12a2z2 + 4z2−4a6 + 8a4−4a2 + 1
Kauffman polynomial (db, data sources) a4z10 + a2z10 + 2a5z9 + 5a3z9 + 3az9 + 3a6z8 + 2a4z8 + 3a2z8 + 4z8 + 3a7z7−a5z7−14a3z7−7az7 + 3z7a−1 + 2a8z6−6a6z6−11a4z6−17a2z6 + z6a−2−13z6 + a9z5−6a7z5−4a5z5 + 14a3z5 + az5−10z5a−1−4a8z4 + 9a6z4 + 24a4z4 + 25a2z4−3z4a−2 + 11z4−3a9z3 + 4a7z3 + 11a5z3 + 2a3z3 + 5az3 + 7z3a−1 + a8z2−10a6z2−21a4z2−15a2z2 + z2a−2−4z2 + 2a9z−2a7z−7a5z−5a3z−3az−za−1 + 4a6 + 8a4 + 4a2 + 1
The A2 invariant −q24−q22−q20−2q18 + 2q16 + 3q12 + 3q10 + 3q6−3q4 + q2−1−q−2 + q−4−q−6 + q−8
The G2 invariant q128−q126 + 3q124−4q122 + 4q120−3q118−q116 + 8q114−15q112 + 21q110−22q108 + 14q106−2q104−18q102 + 39q100−53q98 + 53q96−41q94 + 9q92 + 23q90−57q88 + 78q86−84q84 + 66q82−32q80−20q78 + 63q76−91q74 + 91q72−62q70 + 11q68 + 39q66−70q64 + 68q62−27q60−25q58 + 76q56−83q54 + 50q52 + 20q50−89q48 + 143q46−137q44 + 89q42−q40−84q38 + 151q36−164q34 + 128q32−53q30−32q28 + 98q26−124q24 + 104q22−46q20−23q18 + 69q16−85q14 + 50q12 + 10q10−72q8 + 103q6−93q4 + 38q2 + 35−100q−2 + 128q−4−112q−6 + 63q−8 + 2q−10−61q−12 + 93q−14−91q−16 + 68q−18−28q−20−8q−22 + 31q−24−40q−26 + 36q−28−24q−30 + 12q−32 + q−34−7q−36 + 7q−38−7q−40 + 4q−42−2q−44 + q−46
Further Quantum Invariants
K11a221 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["K11a221"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t4 + 5t3−10t2 + 15t−17 + 15t−1−10t−2 + 5t−3−t−4
In[5]:= Conway[K][z]
Out[5]= −z8−3z6 + 4z2 + 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]= { 79, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q3−3q2 + 5q−8 + 11q−1−12q−2 + 13q−3−10q−4 + 8q−5−5q−6 + 2q−7−q−8
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −a2z8 + 2a4z6−6a2z6 + z6−a6z4 + 10a4z4−13a2z4 + 4z4−4a6z2 + 16a4z2−12a2z2 + 4z2−4a6 + 8a4−4a2 + 1
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= a4z10 + a2z10 + 2a5z9 + 5a3z9 + 3az9 + 3a6z8 + 2a4z8 + 3a2z8 + 4z8 + 3a7z7−a5z7−14a3z7−7az7 + 3z7a−1 + 2a8z6−6a6z6−11a4z6−17a2z6 + z6a−2−13z6 + a9z5−6a7z5−4a5z5 + 14a3z5 + az5−10z5a−1−4a8z4 + 9a6z4 + 24a4z4 + 25a2z4−3z4a−2 + 11z4−3a9z3 + 4a7z3 + 11a5z3 + 2a3z3 + 5az3 + 7z3a−1 + a8z2−10a6z2−21a4z2−15a2z2 + z2a−2−4z2 + 2a9z−2a7z−7a5z−5a3z−3az−za−1 + 4a6 + 8a4 + 4a2 + 1

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

Same Alexander/Conway Polynomial: {K11a259,}

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["K11a221"];
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 + 5t3−10t2 + 15t−17 + 15t−1−10t−2 + 5t−3−t−4, q3−3q2 + 5q−8 + 11q−1−12q−2 + 13q−3−10q−4 + 8q−5−5q−6 + 2q−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]= {K11a259,}
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: (4, -8)

[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 K11a221. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -7-6-5-4-3-2-101234χ
7           11
5          2 -2
3         31 2
1        52  -3
-1       63   3
-3      76    -1
-5     65     1
-7    47      3
-9   46       -2
-11  14        3
-13 14         -3
-15 1          1
-171           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −3 i = −1
r = −7 {\mathbb Z}
r = −6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r = 1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 4 {\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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K11a220

K11a222

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