K11a223

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K11a222

K11a224

Contents

Image:K11a223.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a223's page at Knotilus!

Visit K11a223's page at the original Knot Atlas!

[edit K11a223 Quick Notes]


[edit K11a223 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X18,6,19,5 X20,8,21,7 X16,10,17,9 X2,12,3,11 X22,13,1,14 X8,16,9,15 X10,18,11,17 X6,20,7,19 X14,21,15,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -8, 5, -9, 6, -2, 7, -11, 8, -5, 9, -3, 10, -4, 11, -7
Dowker-Thistlethwaite code 4 12 18 20 16 2 22 8 10 6 14
A Braid Representative
Image:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gif
A Morse Link Presentation Image:K11a223_ML.gif

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 4
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a223/ThurstonBennequinNumber
Hyperbolic Volume 12.6981
A-Polynomial See Data:K11a223/A-polynomial

[edit Notes for K11a223's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant -6

[edit Notes for K11a223's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t4 + 5t3−10t2 + 14t−15 + 14t−1−10t−2 + 5t−3−t−4
Conway polynomial −z8−3z6 + 3z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 75, 6 }
Jones polynomial −q12 + 3q11−5q10 + 8q9−11q8 + 11q7−11q6 + 10q5−7q4 + 5q3−2q2 + q
HOMFLY-PT polynomial (db, data sources) −z8a−6 + z6a−4−6z6a−6 + 2z6a−8 + 5z4a−4−13z4a−6 + 9z4a−8−z4a−10 + 8z2a−4−13z2a−6 + 11z2a−8−3z2a−10 + 4a−4−5a−6 + 3a−8−a−10
Kauffman polynomial (db, data sources) z10a−6 + z10a−8 + 2z9a−5 + 6z9a−7 + 4z9a−9 + z8a−4 + z8a−6 + 8z8a−8 + 8z8a−10−10z7a−5−22z7a−7−3z7a−9 + 9z7a−11−6z6a−4−22z6a−6−44z6a−8−21z6a−10 + 7z6a−12 + 15z5a−5 + 15z5a−7−25z5a−9−20z5a−11 + 5z5a−13 + 13z4a−4 + 43z4a−6 + 55z4a−8 + 14z4a−10−8z4a−12 + 3z4a−14−5z3a−5 + 9z3a−7 + 30z3a−9 + 12z3a−11−3z3a−13 + z3a−15−12z2a−4−27z2a−6−22z2a−8−6z2a−10−z2a−14−3za−5−6za−7−6za−9−3za−11 + 4a−4 + 5a−6 + 3a−8 + a−10
The A2 invariant q−4 + 2q−8 + q−10 + 2q−14−2q−16 + 2q−18−2q−20−q−22−2q−26 + 2q−28 + q−32−q−36
The G2 invariant q−22−q−24 + 5q−26−7q−28 + 10q−30−9q−32 + 3q−34 + 15q−36−31q−38 + 47q−40−46q−42 + 27q−44 + 13q−46−57q−48 + 91q−50−90q−52 + 63q−54−4q−56−56q−58 + 93q−60−96q−62 + 63q−64−7q−66−48q−68 + 69q−70−58q−72 + 19q−74 + 30q−76−66q−78 + 71q−80−48q−82−4q−84 + 57q−86−105q−88 + 112q−90−80q−92 + 26q−94 + 44q−96−100q−98 + 121q−100−100q−102 + 47q−104 + 16q−106−71q−108 + 85q−110−57q−112 + 14q−114 + 31q−116−52q−118 + 47q−120−14q−122−29q−124 + 54q−126−61q−128 + 47q−130−10q−132−25q−134 + 48q−136−53q−138 + 47q−140−31q−142 + 8q−144 + 9q−146−27q−148 + 33q−150−32q−152 + 26q−154−13q−156 + 4q−158 + 7q−160−19q−162 + 21q−164−21q−166 + 15q−168−7q−170 + q−172 + 6q−174−11q−176 + 13q−178−10q−180 + 7q−182−2q−184−q−186 + 2q−188−4q−190 + 3q−192−2q−194 + q−196
Further Quantum Invariants
K11a223 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["K11a223"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t4 + 5t3−10t2 + 14t−15 + 14t−1−10t−2 + 5t−3−t−4
In[5]:= Conway[K][z]
Out[5]= −z8−3z6 + 3z2 + 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]= { 75, 6 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q12 + 3q11−5q10 + 8q9−11q8 + 11q7−11q6 + 10q5−7q4 + 5q3−2q2 + q
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z8a−6 + z6a−4−6z6a−6 + 2z6a−8 + 5z4a−4−13z4a−6 + 9z4a−8−z4a−10 + 8z2a−4−13z2a−6 + 11z2a−8−3z2a−10 + 4a−4−5a−6 + 3a−8−a−10
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z10a−6 + z10a−8 + 2z9a−5 + 6z9a−7 + 4z9a−9 + z8a−4 + z8a−6 + 8z8a−8 + 8z8a−10−10z7a−5−22z7a−7−3z7a−9 + 9z7a−11−6z6a−4−22z6a−6−44z6a−8−21z6a−10 + 7z6a−12 + 15z5a−5 + 15z5a−7−25z5a−9−20z5a−11 + 5z5a−13 + 13z4a−4 + 43z4a−6 + 55z4a−8 + 14z4a−10−8z4a−12 + 3z4a−14−5z3a−5 + 9z3a−7 + 30z3a−9 + 12z3a−11−3z3a−13 + z3a−15−12z2a−4−27z2a−6−22z2a−8−6z2a−10−z2a−14−3za−5−6za−7−6za−9−3za−11 + 4a−4 + 5a−6 + 3a−8 + a−10

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

Same Alexander/Conway Polynomial: {K11n148,}

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["K11a223"];
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 + 14t−15 + 14t−1−10t−2 + 5t−3−t−4, −q12 + 3q11−5q10 + 8q9−11q8 + 11q7−11q6 + 10q5−7q4 + 5q3−2q2 + q }
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]= {K11n148,}
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: (3, 6)

[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 = 6 is the signature of K11a223. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -2-10123456789χ
25           1-1
23          2 2
21         31 -2
19        52  3
17       63   -3
15      55    0
13     66     0
11    45      -1
9   36       3
7  24        -2
5 14         3
3 1          -1
11           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 5 i = 7
r = −2 {\mathbb Z}
r = −1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = 0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r = 1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 8 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 9 {\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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K11a222

K11a224

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