K11a139

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K11a138

K11a140

Contents

Image:K11a139.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a139's page at Knotilus!

Visit K11a139's page at the original Knot Atlas!

[edit K11a139 Quick Notes]


[edit K11a139 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X16,6,17,5 X18,8,19,7 X2,10,3,9 X20,11,21,12 X22,13,1,14 X8,16,9,15 X6,18,7,17 X14,19,15,20 X12,21,13,22
Gauss code 1, -5, 2, -1, 3, -9, 4, -8, 5, -2, 6, -11, 7, -10, 8, -3, 9, -4, 10, -6, 11, -7
Dowker-Thistlethwaite code 4 10 16 18 2 20 22 8 6 14 12
A Braid Representative
Image:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart2.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart0.gif
A Morse Link Presentation Image:K11a139_ML.gif

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 4
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a139/ThurstonBennequinNumber
Hyperbolic Volume 13.7305
A-Polynomial See Data:K11a139/A-polynomial

[edit Notes for K11a139's three dimensional invariants]

[edit] Four dimensional invariants

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

[edit Notes for K11a139's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t4 + 5t3−12t2 + 20t−23 + 20t−1−12t−2 + 5t−3−t−4
Conway polynomial −z8−3z6−2z4 + z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 99, 2 }
Jones polynomial −q8 + 3q7−6q6 + 10q5−14q4 + 16q3−15q2 + 14q−10 + 6q−1−3q−2 + q−3
HOMFLY-PT polynomial (db, data sources) −z8a−2−6z6a−2 + 2z6a−4 + z6−14z4a−2 + 9z4a−4−z4a−6 + 4z4−14z2a−2 + 13z2a−4−3z2a−6 + 5z2−4a−2 + 5a−4−2a−6 + 2
Kauffman polynomial (db, data sources) z10a−2 + z10a−4 + 3z9a−1 + 6z9a−3 + 3z9a−5 + 6z8a−2 + 7z8a−4 + 5z8a−6 + 4z8 + 3az7−3z7a−1−10z7a−3 + z7a−5 + 5z7a−7 + a2z6−20z6a−2−21z6a−4−9z6a−6 + 3z6a−8−10z6−9az5−5z5a−1 + 4z5a−3−12z5a−5−11z5a−7 + z5a−9−3a2z4 + 25z4a−2 + 29z4a−4 + 8z4a−6−6z4a−8 + 7z4 + 7az3 + 5z3a−1 + 4z3a−3 + 17z3a−5 + 9z3a−7−2z3a−9 + 2a2z2−19z2a−2−19z2a−4−4z2a−6 + 2z2a−8−4z2−az−2za−1−3za−3−5za−5−3za−7 + 4a−2 + 5a−4 + 2a−6 + 2
The A2 invariant q8−q6 + 2q4−q2−1 + 2q−2−3q−4 + 4q−6−q−8 + q−10 + q−12−2q−14 + 3q−16−q−18−q−24
The G2 invariant q46−2q44 + 5q42−9q40 + 10q38−10q36 + 2q34 + 15q32−34q30 + 55q28−63q26 + 50q24−15q22−44q20 + 112q18−162q16 + 172q14−120q12 + 15q10 + 115q8−220q6 + 271q4−231q2 + 111 + 47q−2−192q−4 + 253q−6−214q−8 + 98q−10 + 53q−12−158q−14 + 179q−16−114q−18−25q−20 + 168q−22−258q−24 + 232q−26−109q−28−77q−30 + 265q−32−369q−34 + 357q−36−226q−38 + 23q−40 + 189q−42−339q−44 + 362q−46−257q−48 + 90q−50 + 97q−52−207q−54 + 215q−56−124q−58−9q−60 + 126q−62−179q−64 + 133q−66−18q−68−116q−70 + 219q−72−235q−74 + 174q−76−60q−78−77q−80 + 175q−82−225q−84 + 203q−86−128q−88 + 34q−90 + 55q−92−112q−94 + 129q−96−112q−98 + 72q−100−24q−102−18q−104 + 39q−106−46q−108 + 39q−110−24q−112 + 12q−114 + q−116−7q−118 + 7q−120−7q−122 + 4q−124−2q−126 + q−128
Further Quantum Invariants
K11a139 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["K11a139"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t4 + 5t3−12t2 + 20t−23 + 20t−1−12t−2 + 5t−3−t−4
In[5]:= Conway[K][z]
Out[5]= −z8−3z6−2z4 + 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]= { 99, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q8 + 3q7−6q6 + 10q5−14q4 + 16q3−15q2 + 14q−10 + 6q−1−3q−2 + q−3
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z8a−2−6z6a−2 + 2z6a−4 + z6−14z4a−2 + 9z4a−4−z4a−6 + 4z4−14z2a−2 + 13z2a−4−3z2a−6 + 5z2−4a−2 + 5a−4−2a−6 + 2
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z10a−2 + z10a−4 + 3z9a−1 + 6z9a−3 + 3z9a−5 + 6z8a−2 + 7z8a−4 + 5z8a−6 + 4z8 + 3az7−3z7a−1−10z7a−3 + z7a−5 + 5z7a−7 + a2z6−20z6a−2−21z6a−4−9z6a−6 + 3z6a−8−10z6−9az5−5z5a−1 + 4z5a−3−12z5a−5−11z5a−7 + z5a−9−3a2z4 + 25z4a−2 + 29z4a−4 + 8z4a−6−6z4a−8 + 7z4 + 7az3 + 5z3a−1 + 4z3a−3 + 17z3a−5 + 9z3a−7−2z3a−9 + 2a2z2−19z2a−2−19z2a−4−4z2a−6 + 2z2a−8−4z2−az−2za−1−3za−3−5za−5−3za−7 + 4a−2 + 5a−4 + 2a−6 + 2

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

Same Alexander/Conway Polynomial: {K11a57, K11a108, K11a231,}

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["K11a139"];
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 + 5t3−12t2 + 20t−23 + 20t−1−12t−2 + 5t−3−t−4, −q8 + 3q7−6q6 + 10q5−14q4 + 16q3−15q2 + 14q−10 + 6q−1−3q−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]= {K11a57, K11a108, K11a231,}
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: (1, 3)

[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 K11a139. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-101234567χ
17           1-1
15          2 2
13         41 -3
11        62  4
9       84   -4
7      86    2
5     78     1
3    78      -1
1   48       4
-1  26        -4
-3 14         3
-5 2          -2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r = 1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = 3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = 4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11a138

K11a140

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