K11a108

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K11a107

K11a109

Contents

Image:K11a108.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a108's page at Knotilus!

Visit K11a108's page at the original Knot Atlas!

[edit K11a108 Quick Notes]


[edit K11a108 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 X18,15,19,16 X6,18,7,17 X8,19,9,20 X12,21,13,22
Gauss code 1, -5, 2, -1, 3, -9, 4, -10, 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 6 8 12
A Braid Representative
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gif
A Morse Link Presentation Image:K11a108_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:K11a108/ThurstonBennequinNumber
Hyperbolic Volume 14.2368
A-Polynomial See Data:K11a108/A-polynomial

[edit Notes for K11a108'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 K11a108'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 −q4 + 3q3−6q2 + 10q−13 + 16q−1−15q−2 + 14q−3−11q−4 + 6q−5−3q−6 + q−7
HOMFLY-PT polynomial (db, data sources) −a2z8 + a4z6−6a2z6 + 2z6 + 4a4z4−14a2z4−z4a−2 + 9z4 + 5a4z2−14a2z2−3z2a−2 + 13z2 + a4−4a2−2a−2 + 6
Kauffman polynomial (db, data sources) a2z10 + z10 + 4a3z9 + 7az9 + 3z9a−1 + 7a4z8 + 11a2z8 + 3z8a−2 + 7z8 + 7a5z7 + a3z7−14az7−7z7a−1 + z7a−3 + 5a6z6−11a4z6−38a2z6−12z6a−2−34z6 + 3a7z5−9a5z5−17a3z5−4az5−3z5a−1−4z5a−3 + a8z4−4a6z4 + 9a4z4 + 41a2z4 + 15z4a−2 + 42z4−3a7z3 + 5a5z3 + 15a3z3 + 14az3 + 12z3a−1 + 5z3a−3−a8z2 + a6z2−5a4z2−23a2z2−7z2a−2−23z2 + a7z + a5z−3a3z−6az−5za−1−2za−3 + a4 + 4a2 + 2a−2 + 6
The A2 invariant q20−q18 + 2q16−2q14−2q12 + q10−3q8 + 4q6−q4 + 2q2 + 2−q−2 + 3q−4−q−6−q−12
The G2 invariant q114−2q112 + 4q110−6q108 + 5q106−4q104−2q102 + 10q100−18q98 + 26q96−30q94 + 26q92−11q90−11q88 + 42q86−67q84 + 82q82−85q80 + 58q78−10q76−54q74 + 123q72−159q70 + 166q68−121q66 + 33q64 + 71q62−169q60 + 217q58−192q56 + 95q54 + 28q52−133q50 + 183q48−144q46 + 33q44 + 91q42−185q40 + 182q38−89q36−77q34 + 236q32−310q30 + 275q28−132q26−74q24 + 261q22−364q20 + 344q18−214q16 + 19q14 + 174q12−275q10 + 279q8−174q6 + 20q4 + 124q2−200 + 173q−2−59q−4−86q−6 + 209q−8−233q−10 + 159q−12−12q−14−145q−16 + 257q−18−275q−20 + 196q−22−61q−24−85q−26 + 184q−28−207q−30 + 165q−32−82q−34−3q−36 + 61q−38−88q−40 + 76q−42−48q−44 + 18q−46 + 4q−48−15q−50 + 14q−52−11q−54 + 6q−56−2q−58 + q−60
Further Quantum Invariants
K11a108 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["K11a108"];
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]= −q4 + 3q3−6q2 + 10q−13 + 16q−1−15q−2 + 14q−3−11q−4 + 6q−5−3q−6 + q−7
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −a2z8 + a4z6−6a2z6 + 2z6 + 4a4z4−14a2z4−z4a−2 + 9z4 + 5a4z2−14a2z2−3z2a−2 + 13z2 + a4−4a2−2a−2 + 6
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= a2z10 + z10 + 4a3z9 + 7az9 + 3z9a−1 + 7a4z8 + 11a2z8 + 3z8a−2 + 7z8 + 7a5z7 + a3z7−14az7−7z7a−1 + z7a−3 + 5a6z6−11a4z6−38a2z6−12z6a−2−34z6 + 3a7z5−9a5z5−17a3z5−4az5−3z5a−1−4z5a−3 + a8z4−4a6z4 + 9a4z4 + 41a2z4 + 15z4a−2 + 42z4−3a7z3 + 5a5z3 + 15a3z3 + 14az3 + 12z3a−1 + 5z3a−3−a8z2 + a6z2−5a4z2−23a2z2−7z2a−2−23z2 + a7z + a5z−3a3z−6az−5za−1−2za−3 + a4 + 4a2 + 2a−2 + 6

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

Same Alexander/Conway Polynomial: {K11a57, K11a139, 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["K11a108"];
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, −q4 + 3q3−6q2 + 10q−13 + 16q−1−15q−2 + 14q−3−11q−4 + 6q−5−3q−6 + q−7 }
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, K11a139, 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, 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 = -2 is the signature of K11a108. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-1012345χ
9           1-1
7          2 2
5         41 -3
3        62  4
1       74   -3
-1      96    3
-3     78     1
-5    78      -1
-7   47       3
-9  27        -5
-11 14         3
-13 2          -2
-151           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −3 i = −1
r = −6 {\mathbb Z}
r = −5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = −2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = −1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r = 1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11a107

K11a109

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