K11a80

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K11a79

K11a81

Contents

Image:K11a80.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a80's page at Knotilus!

Visit K11a80's page at the original Knot Atlas!

[edit K11a80 Quick Notes]


[edit K11a80 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X22,10,1,9 X2,11,3,12 X18,14,19,13 X20,15,21,16 X6,17,7,18 X8,20,9,19 X16,21,17,22
Gauss code 1, -6, 2, -1, 3, -9, 4, -10, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, -8, 11, -5
Dowker-Thistlethwaite code 4 10 12 14 22 2 18 20 6 8 16
A Braid Representative
Image:BraidPart1.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:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart4.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gif
A Morse Link Presentation Image:K11a80_ML.gif

[edit] Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 4
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a80/ThurstonBennequinNumber
Hyperbolic Volume 16.5161
A-Polynomial See Data:K11a80/A-polynomial

[edit Notes for K11a80's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 0

[edit Notes for K11a80's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t4−6t3 + 16t2−28t + 35−28t−1 + 16t−2−6t−3 + t−4
Conway polynomial z8 + 2z6−2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 137, 0 }
Jones polynomial −q5 + 4q4−8q3 + 14q2−19q + 22−22q−1 + 19q−2−14q−3 + 9q−4−4q−5 + q−6
HOMFLY-PT polynomial (db, data sources) z8−2a2z6−z6a−2 + 5z6 + a4z4−7a2z4−3z4a−2 + 9z4 + 2a4z2−7a2z2−2z2a−2 + 5z2 + a4−a2 + a−2
Kauffman polynomial (db, data sources) 2a2z10 + 2z10 + 6a3z9 + 12az9 + 6z9a−1 + 7a4z8 + 12a2z8 + 8z8a−2 + 13z8 + 4a5z7−7a3z7−19az7−z7a−1 + 7z7a−3 + a6z6−16a4z6−38a2z6−9z6a−2 + 4z6a−4−34z6−9a5z5−7a3z5 + az5−12z5a−1−10z5a−3 + z5a−5−2a6z4 + 10a4z4 + 33a2z4−6z4a−4 + 27z4 + 5a5z3 + 8a3z3 + 11az3 + 13z3a−1 + 4z3a−3−z3a−5 + a6z2−4a4z2−11a2z2 + 3z2a−2 + 2z2a−4−5z2−a5z−2a3z−4az−4za−1−za−3 + a4 + a2−a−2
The A2 invariant q18−q16 + 2q12−3q10 + 4q8−q6−q4 + 2q2−5 + 4q−2−3q−4 + 2q−6 + 3q−8−2q−10 + 2q−12−q−14
The G2 invariant q94−3q92 + 8q90−16q88 + 22q86−24q84 + 12q82 + 20q80−66q78 + 122q76−163q74 + 153q72−74q70−81q68 + 278q66−435q64 + 489q62−371q60 + 80q58 + 297q56−636q54 + 791q52−674q50 + 309q48 + 174q46−588q44 + 763q42−624q40 + 243q38 + 218q36−548q34 + 593q32−336q30−117q28 + 568q26−804q24 + 717q22−313q20−267q18 + 803q16−1097q14 + 1025q12−605q10−21q8 + 624q6−994q4 + 998q2−656 + 120q−2 + 386q−4−666q−6 + 614q−8−284q−10−153q−12 + 499q−14−586q−16 + 384q−18 + 11q−20−425q−22 + 690q−24−694q−26 + 460q−28−82q−30−296q−32 + 542q−34−600q−36 + 486q−38−256q−40 + 10q−42 + 184q−44−289q−46 + 294q−48−230q−50 + 132q−52−31q−54−46q−56 + 86q−58−95q−60 + 77q−62−46q−64 + 20q−66 + 3q−68−14q−70 + 15q−72−13q−74 + 7q−76−3q−78 + q−80
Further Quantum Invariants
K11a80 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["K11a80"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t4−6t3 + 16t2−28t + 35−28t−1 + 16t−2−6t−3 + t−4
In[5]:= Conway[K][z]
Out[5]= z8 + 2z6−2z2 + 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]= { 137, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q5 + 4q4−8q3 + 14q2−19q + 22−22q−1 + 19q−2−14q−3 + 9q−4−4q−5 + q−6
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z8−2a2z6−z6a−2 + 5z6 + a4z4−7a2z4−3z4a−2 + 9z4 + 2a4z2−7a2z2−2z2a−2 + 5z2 + a4−a2 + a−2
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 2a2z10 + 2z10 + 6a3z9 + 12az9 + 6z9a−1 + 7a4z8 + 12a2z8 + 8z8a−2 + 13z8 + 4a5z7−7a3z7−19az7−z7a−1 + 7z7a−3 + a6z6−16a4z6−38a2z6−9z6a−2 + 4z6a−4−34z6−9a5z5−7a3z5 + az5−12z5a−1−10z5a−3 + z5a−5−2a6z4 + 10a4z4 + 33a2z4−6z4a−4 + 27z4 + 5a5z3 + 8a3z3 + 11az3 + 13z3a−1 + 4z3a−3−z3a−5 + a6z2−4a4z2−11a2z2 + 3z2a−2 + 2z2a−4−5z2−a5z−2a3z−4az−4za−1−za−3 + a4 + a2−a−2

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a270,}

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["K11a80"];
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]= { t4−6t3 + 16t2−28t + 35−28t−1 + 16t−2−6t−3 + t−4, −q5 + 4q4−8q3 + 14q2−19q + 22−22q−1 + 19q−2−14q−3 + 9q−4−4q−5 + q−6 }
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]= {K11a270,}

[edit] Vassiliev invariants

V2 and V3: (-2, 1)

[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 = 0 is the signature of K11a80. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-1012345χ
11           1-1
9          3 3
7         51 -4
5        93  6
3       105   -5
1      129    3
-1     1111     0
-3    811      -3
-5   611       5
-7  38        -5
-9 16         5
-11 3          -3
-131           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −1 i = 1
r = −6 {\mathbb Z}
r = −5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = −2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = −1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r = 0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r = 1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r = 2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r = 3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 5 {\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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K11a79

K11a81

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