K11a171

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K11a170

K11a172

Contents

Image:K11a171.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a171's page at Knotilus!

Visit K11a171's page at the original Knot Atlas!

[edit K11a171 Quick Notes]


[edit K11a171 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X18,5,19,6 X20,8,21,7 X2,10,3,9 X22,11,1,12 X6,14,7,13 X8,15,9,16 X12,18,13,17 X14,19,15,20 X16,22,17,21
Gauss code 1, -5, 2, -1, 3, -7, 4, -8, 5, -2, 6, -9, 7, -10, 8, -11, 9, -3, 10, -4, 11, -6
Dowker-Thistlethwaite code 4 10 18 20 2 22 6 8 12 14 16
A Braid Representative
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gif
A Morse Link Presentation Image:K11a171_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:K11a171/ThurstonBennequinNumber
Hyperbolic Volume 18.6712
A-Polynomial See Data:K11a171/A-polynomial

[edit Notes for K11a171'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 K11a171's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t4 + 7t3−21t2 + 39t−47 + 39t−1−21t−2 + 7t−3−t−4
Conway polynomial −z8−z6 + z4 + 2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 183, 2 }
Jones polynomial −q8 + 5q7−12q6 + 19q5−26q4 + 30q3−29q2 + 26q−18 + 11q−1−5q−2 + q−3
HOMFLY-PT polynomial (db, data sources) −z8a−2−4z6a−2 + 2z6a−4 + z6−5z4a−2 + 5z4a−4−z4a−6 + 2z4 + z2a−2 + 2z2a−4−z2a−6 + 4a−2−2a−4−1
Kauffman polynomial (db, data sources) 3z10a−2 + 3z10a−4 + 9z9a−1 + 20z9a−3 + 11z9a−5 + 23z8a−2 + 29z8a−4 + 16z8a−6 + 10z8 + 5az7−7z7a−1−23z7a−3 + z7a−5 + 12z7a−7 + a2z6−65z6a−2−71z6a−4−23z6a−6 + 5z6a−8−21z6−9az5−15z5a−1−18z5a−3−28z5a−5−15z5a−7 + z5a−9−a2z4 + 47z4a−2 + 47z4a−4 + 12z4a−6−3z4a−8 + 14z4 + 4az3 + 14z3a−1 + 21z3a−3 + 17z3a−5 + 6z3a−7−6z2a−2−8z2a−4−4z2a−6−2z2−za−1−za−3−za−5−za−7−4a−2−2a−4−1
The A2 invariant q8−3q6 + 3q4−3q2−1 + 6q−2−4q−4 + 8q−6−2q−8 + q−10 + q−12−6q−14 + 4q−16−3q−18 + 2q−22−q−24
The G2 invariant q46−4q44 + 11q42−24q40 + 36q38−43q36 + 30q34 + 20q32−100q30 + 210q28−303q26 + 316q24−204q22−78q20 + 476q18−859q16 + 1062q14−921q12 + 376q10 + 457q8−1316q6 + 1850q4−1784q2 + 1060 + 97q−2−1270q−4 + 1969q−6−1893q−8 + 1078q−10 + 148q−12−1230q−14 + 1686q−16−1303q−18 + 249q−20 + 1016q−22−1900q−24 + 2001q−26−1197q−28−222q−30 + 1715q−32−2695q−34 + 2772q−36−1870q−38 + 304q−40 + 1366q−42−2538q−44 + 2797q−46−2076q−48 + 684q−50 + 801q−52−1800q−54 + 1931q−56−1212q−58 + 9q−60 + 1116q−62−1648q−64 + 1352q−66−394q−68−811q−70 + 1726q−72−1967q−74 + 1468q−76−447q−78−674q−80 + 1480q−82−1735q−84 + 1427q−86−747q−88−13q−90 + 601q−92−883q−94 + 847q−96−591q−98 + 270q−100 + 17q−102−196q−104 + 252q−106−230q−108 + 152q−110−71q−112 + 14q−114 + 22q−116−31q−118 + 28q−120−20q−122 + 10q−124−4q−126 + q−128
Further Quantum Invariants
K11a171 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["K11a171"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t4 + 7t3−21t2 + 39t−47 + 39t−1−21t−2 + 7t−3−t−4
In[5]:= Conway[K][z]
Out[5]= −z8−z6 + 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]= { 183, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q8 + 5q7−12q6 + 19q5−26q4 + 30q3−29q2 + 26q−18 + 11q−1−5q−2 + q−3
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z8a−2−4z6a−2 + 2z6a−4 + z6−5z4a−2 + 5z4a−4−z4a−6 + 2z4 + z2a−2 + 2z2a−4−z2a−6 + 4a−2−2a−4−1
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 3z10a−2 + 3z10a−4 + 9z9a−1 + 20z9a−3 + 11z9a−5 + 23z8a−2 + 29z8a−4 + 16z8a−6 + 10z8 + 5az7−7z7a−1−23z7a−3 + z7a−5 + 12z7a−7 + a2z6−65z6a−2−71z6a−4−23z6a−6 + 5z6a−8−21z6−9az5−15z5a−1−18z5a−3−28z5a−5−15z5a−7 + z5a−9−a2z4 + 47z4a−2 + 47z4a−4 + 12z4a−6−3z4a−8 + 14z4 + 4az3 + 14z3a−1 + 21z3a−3 + 17z3a−5 + 6z3a−7−6z2a−2−8z2a−4−4z2a−6−2z2−za−1−za−3−za−5−za−7−4a−2−2a−4−1

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

Same Alexander/Conway Polynomial: {}

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["K11a171"];
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 + 7t3−21t2 + 39t−47 + 39t−1−21t−2 + 7t−3−t−4, −q8 + 5q7−12q6 + 19q5−26q4 + 30q3−29q2 + 26q−18 + 11q−1−5q−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]= {}
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 K11a171. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-101234567χ
17           1-1
15          4 4
13         81 -7
11        114  7
9       158   -7
7      1511    4
5     1415     1
3    1215      -3
1   715       8
-1  411        -7
-3 17         6
-5 4          -4
-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}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = −1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 0 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r = 1 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r = 2 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r = 3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r = 4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r = 5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = 6 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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K11a170

K11a172

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