K11a86

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K11a85

K11a87

Contents

Image:K11a86.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a86's page at Knotilus!

Visit K11a86's page at the original Knot Atlas!

[edit K11a86 Quick Notes]


[edit K11a86 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X16,8,17,7 X18,10,19,9 X2,11,3,12 X20,13,21,14 X6,16,7,15 X8,18,9,17 X22,19,1,20 X14,21,15,22
Gauss code 1, -6, 2, -1, 3, -8, 4, -9, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, -7, 11, -10
Dowker-Thistlethwaite code 4 10 12 16 18 2 20 6 8 22 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:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gif
A Morse Link Presentation Image:K11a86_ML.gif

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a86/ThurstonBennequinNumber
Hyperbolic Volume 13.5574
A-Polynomial See Data:K11a86/A-polynomial

[edit Notes for K11a86'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 K11a86's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t4 + 5t3−12t2 + 18t−19 + 18t−1−12t−2 + 5t−3−t−4
Conway polynomial −z8−3z6−2z4−z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 91, 2 }
Jones polynomial q7−3q6 + 6q5−10q4 + 13q3−14q2 + 14q−12 + 9q−1−5q−2 + 3q−3−q−4
HOMFLY-PT polynomial (db, data sources) −z8a−2−6z6a−2 + z6a−4 + 2z6−a2z4−14z4a−2 + 4z4a−4 + 9z4−3a2z2−15z2a−2 + 5z2a−4 + 12z2−a2−5a−2 + 2a−4 + 5
Kauffman polynomial (db, data sources) z10a−2 + z10 + 3az9 + 7z9a−1 + 4z9a−3 + 3a2z8 + 9z8a−2 + 6z8a−4 + 6z8 + a3z7−9az7−20z7a−1−4z7a−3 + 6z7a−5−13a2z6−36z6a−2−9z6a−4 + 5z6a−6−35z6−4a3z5 + 3az5 + 14z5a−1−2z5a−3−6z5a−5 + 3z5a−7 + 17a2z4 + 45z4a−2 + 7z4a−4−5z4a−6 + z4a−8 + 49z4 + 4a3z3 + 5az3−2z3a−1 + z3a−3 + z3a−5−3z3a−7−8a2z2−26z2a−2−5z2a−4 + 2z2a−6−z2a−8−26z2−a3z−2az−za−1 + za−5 + za−7 + a2 + 5a−2 + 2a−4 + 5
The A2 invariant −q12 + q8 + 3q4−q2 + 1 + q−2−2q−4 + 3q−6−3q−8 + q−10−q−12−q−14 + 2q−16−q−18 + q−20
The G2 invariant q60−2q58 + 5q56−9q54 + 10q52−11q50 + 3q48 + 14q46−35q44 + 57q42−66q40 + 48q38−8q36−54q34 + 115q32−153q30 + 144q28−77q26−25q24 + 131q22−196q20 + 197q18−123q16 + 13q14 + 97q12−163q10 + 159q8−78q6−23q4 + 116q2−144 + 96q−2−2q−4−111q−6 + 188q−8−199q−10 + 135q−12−10q−14−126q−16 + 233q−18−263q−20 + 201q−22−81q−24−64q−26 + 176q−28−222q−30 + 186q−32−84q−34−29q−36 + 115q−38−137q−40 + 82q−42−82q−46 + 119q−48−98q−50 + 32q−52 + 51q−54−118q−56 + 147q−58−124q−60 + 63q−62 + 7q−64−72q−66 + 110q−68−116q−70 + 101q−72−58q−74 + 14q−76 + 30q−78−62q−80 + 71q−82−66q−84 + 48q−86−20q−88−4q−90 + 22q−92−30q−94 + 28q−96−20q−98 + 11q−100−2q−102−4q−104 + 5q−106−6q−108 + 4q−110−2q−112 + q−114
Further Quantum Invariants
K11a86 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["K11a86"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t4 + 5t3−12t2 + 18t−19 + 18t−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]= { 91, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q7−3q6 + 6q5−10q4 + 13q3−14q2 + 14q−12 + 9q−1−5q−2 + 3q−3−q−4
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z8a−2−6z6a−2 + z6a−4 + 2z6−a2z4−14z4a−2 + 4z4a−4 + 9z4−3a2z2−15z2a−2 + 5z2a−4 + 12z2−a2−5a−2 + 2a−4 + 5
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z10a−2 + z10 + 3az9 + 7z9a−1 + 4z9a−3 + 3a2z8 + 9z8a−2 + 6z8a−4 + 6z8 + a3z7−9az7−20z7a−1−4z7a−3 + 6z7a−5−13a2z6−36z6a−2−9z6a−4 + 5z6a−6−35z6−4a3z5 + 3az5 + 14z5a−1−2z5a−3−6z5a−5 + 3z5a−7 + 17a2z4 + 45z4a−2 + 7z4a−4−5z4a−6 + z4a−8 + 49z4 + 4a3z3 + 5az3−2z3a−1 + z3a−3 + z3a−5−3z3a−7−8a2z2−26z2a−2−5z2a−4 + 2z2a−6−z2a−8−26z2−a3z−2az−za−1 + za−5 + za−7 + a2 + 5a−2 + 2a−4 + 5

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a205,}

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["K11a86"];
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 + 18t−19 + 18t−1−12t−2 + 5t−3−t−4, q7−3q6 + 6q5−10q4 + 13q3−14q2 + 14q−12 + 9q−1−5q−2 + 3q−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]= {}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11a205,}

[edit] Vassiliev invariants

V2 and V3: (-1, -2)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
−4 −16 8 \frac{34}{3} \frac{62}{3} 64 \frac{608}{3} \frac{224}{3} 80 -\frac{32}{3} 128 -\frac{136}{3} -\frac{248}{3} \frac{13169}{30} \frac{714}{5} \frac{898}{45} \frac{1999}{18} -\frac{2191}{30}

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

[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 K11a86. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-10123456χ
15           11
13          2 -2
11         41 3
9        62  -4
7       74   3
5      76    -1
3     77     0
1    68      2
-1   36       -3
-3  26        4
-5 13         -2
-7 2          2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r = 1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11a85

K11a87

Retrieved from "http://katlas.math.toronto.edu/wiki/K11a86"
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