K11a109

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K11a108

K11a110

Contents

Image:K11a109.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a109's page at Knotilus!

Visit K11a109's page at the original Knot Atlas!

[edit K11a109 Quick Notes]


[edit K11a109 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,5,15,6 X16,8,17,7 X2,10,3,9 X20,11,21,12 X22,13,1,14 X18,16,19,15 X8,18,9,17 X6,19,7,20 X12,21,13,22
Gauss code 1, -5, 2, -1, 3, -10, 4, -9, 5, -2, 6, -11, 7, -3, 8, -4, 9, -8, 10, -6, 11, -7
Dowker-Thistlethwaite code 4 10 14 16 2 20 22 18 8 6 12
A Braid Representative
Image:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gif
A Morse Link Presentation Image:K11a109_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:K11a109/ThurstonBennequinNumber
Hyperbolic Volume 15.1191
A-Polynomial See Data:K11a109/A-polynomial

[edit Notes for K11a109's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [2,4]
Rasmussen s-Invariant 0

[edit Notes for K11a109's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t4−5t3 + 14t2−24t + 29−24t−1 + 14t−2−5t−3 + t−4
Conway polynomial z8 + 3z6 + 4z4 + 3z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 117, 0 }
Jones polynomial q6−4q5 + 7q4−12q3 + 17q2−18q + 19−16q−1 + 12q−2−7q−3 + 3q−4−q−5
HOMFLY-PT polynomial (db, data sources) z8−a2z6−2z6a−2 + 6z6−4a2z4−8z4a−2 + z4a−4 + 15z4−6a2z2−10z2a−2 + 2z2a−4 + 17z2−3a2−3a−2 + 7
Kauffman polynomial (db, data sources) z10a−2 + z10 + 4az9 + 8z9a−1 + 4z9a−3 + 6a2z8 + 13z8a−2 + 6z8a−4 + 13z8 + 5a3z7 + az7−9z7a−1−z7a−3 + 4z7a−5 + 3a4z6−9a2z6−42z6a−2−16z6a−4 + z6a−6−37z6 + a5z5−7a3z5−11az5−10z5a−1−18z5a−3−11z5a−5−5a4z4 + 9a2z4 + 46z4a−2 + 13z4a−4−2z4a−6 + 45z4−2a5z3 + 3a3z3 + 13az3 + 20z3a−1 + 19z3a−3 + 7z3a−5 + 2a4z2−8a2z2−22z2a−2−5z2a−4−27z2 + a5z−a3z−6az−8za−1−4za−3 + 3a2 + 3a−2 + 7
The A2 invariant −q14 + q12−3q10 + q8 + q6−2q4 + 5q2−2 + 4q−2 + q−4 + 3q−8−4q−10−q−14−q−16 + q−18
The G2 invariant q80−2q78 + 5q76−8q74 + 9q72−8q70 + q68 + 13q66−29q64 + 47q62−59q60 + 53q58−30q56−20q54 + 87q52−151q50 + 190q48−184q46 + 111q44 + 17q42−178q40 + 321q38−383q36 + 329q34−167q32−75q30 + 296q28−427q26 + 416q24−248q22−q20 + 233q18−340q16 + 281q14−81q12−170q10 + 352q8−375q6 + 215q4 + 84q2−392 + 602q−2−592q−4 + 372q−6−11q−8−369q−10 + 626q−12−669q−14 + 501q−16−172q−18−178q−20 + 435q−22−489q−24 + 346q−26−84q−28−187q−30 + 334q−32−310q−34 + 123q−36 + 138q−38−351q−40 + 438q−42−346q−44 + 100q−46 + 172q−48−392q−50 + 469q−52−393q−54 + 203q−56 + 24q−58−208q−60 + 303q−62−291q−64 + 199q−66−76q−68−33q−70 + 95q−72−115q−74 + 96q−76−55q−78 + 22q−80 + 7q−82−18q−84 + 17q−86−14q−88 + 7q−90−3q−92 + q−94
Further Quantum Invariants
K11a109 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["K11a109"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t4−5t3 + 14t2−24t + 29−24t−1 + 14t−2−5t−3 + t−4
In[5]:= Conway[K][z]
Out[5]= z8 + 3z6 + 4z4 + 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]= { 117, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q6−4q5 + 7q4−12q3 + 17q2−18q + 19−16q−1 + 12q−2−7q−3 + 3q−4−q−5
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z8−a2z6−2z6a−2 + 6z6−4a2z4−8z4a−2 + z4a−4 + 15z4−6a2z2−10z2a−2 + 2z2a−4 + 17z2−3a2−3a−2 + 7
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z10a−2 + z10 + 4az9 + 8z9a−1 + 4z9a−3 + 6a2z8 + 13z8a−2 + 6z8a−4 + 13z8 + 5a3z7 + az7−9z7a−1−z7a−3 + 4z7a−5 + 3a4z6−9a2z6−42z6a−2−16z6a−4 + z6a−6−37z6 + a5z5−7a3z5−11az5−10z5a−1−18z5a−3−11z5a−5−5a4z4 + 9a2z4 + 46z4a−2 + 13z4a−4−2z4a−6 + 45z4−2a5z3 + 3a3z3 + 13az3 + 20z3a−1 + 19z3a−3 + 7z3a−5 + 2a4z2−8a2z2−22z2a−2−5z2a−4−27z2 + a5z−a3z−6az−8za−1−4za−3 + 3a2 + 3a−2 + 7

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

Same Alexander/Conway Polynomial: {K11a44, K11a47,}

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["K11a109"];
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 + 14t2−24t + 29−24t−1 + 14t−2−5t−3 + t−4, q6−4q5 + 7q4−12q3 + 17q2−18q + 19−16q−1 + 12q−2−7q−3 + 3q−4−q−5 }
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]= {K11a44, K11a47,}
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, 0)

[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 K11a109. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-10123456χ
13           11
11          3 -3
9         41 3
7        83  -5
5       94   5
3      98    -1
1     109     1
-1    710      3
-3   59       -4
-5  27        5
-7 15         -4
-9 2          2
-111           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −1 i = 1
r = −5 {\mathbb Z}
r = −4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = −1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r = 1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r = 2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r = 3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = 4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11a108

K11a110

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