K11a57

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K11a56

K11a58

Contents

Image:K11a57.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a57's page at Knotilus!

Visit K11a57's page at the original Knot Atlas!

[edit K11a57 Quick Notes]


[edit K11a57 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X8493 X16,6,17,5 X2837 X20,9,21,10 X22,11,1,12 X18,13,19,14 X6,16,7,15 X12,17,13,18 X14,19,15,20 X10,21,11,22
Gauss code 1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -9, 7, -10, 8, -3, 9, -7, 10, -5, 11, -6
Dowker-Thistlethwaite code 4 8 16 2 20 22 18 6 12 14 10
A Braid Representative
Image:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart2.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart0.gif
A Morse Link Presentation Image:K11a57_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:K11a57/ThurstonBennequinNumber
Hyperbolic Volume 14.7354
A-Polynomial See Data:K11a57/A-polynomial

[edit Notes for K11a57's three dimensional invariants]

[edit] Four dimensional invariants

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

[edit Notes for K11a57'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) \left\{t^2-t+1\right\}
Determinant and Signature { 99, -2 }
Jones polynomial −q4 + 2q3−5q2 + 10q−12 + 16q−1−16q−2 + 14q−3−12q−4 + 7q−5−3q−6 + q−7
HOMFLY-PT polynomial (db, data sources) −a2z8 + a4z6−6a2z6 + 2z6 + 4a4z4−15a2z4−z4a−2 + 10z4 + 6a4z2−19a2z2−4z2a−2 + 18z2 + 3a4−10a2−4a−2 + 12
Kauffman polynomial (db, data sources) a2z10 + z10 + 4a3z9 + 6az9 + 2z9a−1 + 8a4z8 + 10a2z8 + 2z8a−2 + 4z8 + 9a5z7 + 4a3z7−9az7−3z7a−1 + z7a−3 + 6a6z6−12a4z6−31a2z6−8z6a−2−21z6 + 3a7z5−16a5z5−29a3z5−14az5−9z5a−1−5z5a−3 + a8z4−6a6z4 + 6a4z4 + 30a2z4 + 11z4a−2 + 28z4−2a7z3 + 16a5z3 + 35a3z3 + 29az3 + 20z3a−1 + 8z3a−3−a8z2 + 3a6z2−2a4z2−22a2z2−8z2a−2−24z2−6a5z−16a3z−17az−11za−1−4za−3 + 3a4 + 10a2 + 4a−2 + 12
The A2 invariant q20−q18 + 3q16−q14−q12−6q8 + q6−3q4 + 4q2 + 5 + 2q−2 + 4q−4−2q−6−q−8−q−10−q−12
The G2 invariant q114−2q112 + 4q110−6q108 + 6q106−5q104 + 9q100−19q98 + 29q96−38q94 + 35q92−22q90−5q88 + 49q86−87q84 + 115q82−119q80 + 82q78−16q76−81q74 + 178q72−232q70 + 223q68−133q66−4q64 + 156q62−252q60 + 268q58−177q56 + 20q54 + 136q52−227q50 + 204q48−65q46−111q44 + 242q42−274q40 + 170q38 + 19q36−243q34 + 375q32−397q30 + 265q28−43q26−210q24 + 377q22−414q20 + 315q18−126q16−99q14 + 262q12−302q10 + 229q8−54q6−119q4 + 242q2−219 + 99q−2 + 86q−4−233q−6 + 304q−8−242q−10 + 84q−12 + 106q−14−243q−16 + 304q−18−249q−20 + 121q−22 + 17q−24−131q−26 + 174q−28−164q−30 + 104q−32−36q−34−21q−36 + 51q−38−59q−40 + 45q−42−26q−44 + 8q−46 + 2q−48−9q−50 + 7q−52−6q−54 + 4q−56−q−58 + q−60
Further Quantum Invariants
K11a57 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["K11a57"];
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]= \left\{t^2-t+1\right\}
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 + 2q3−5q2 + 10q−12 + 16q−1−16q−2 + 14q−3−12q−4 + 7q−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 + 4a4z4−15a2z4−z4a−2 + 10z4 + 6a4z2−19a2z2−4z2a−2 + 18z2 + 3a4−10a2−4a−2 + 12
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= a2z10 + z10 + 4a3z9 + 6az9 + 2z9a−1 + 8a4z8 + 10a2z8 + 2z8a−2 + 4z8 + 9a5z7 + 4a3z7−9az7−3z7a−1 + z7a−3 + 6a6z6−12a4z6−31a2z6−8z6a−2−21z6 + 3a7z5−16a5z5−29a3z5−14az5−9z5a−1−5z5a−3 + a8z4−6a6z4 + 6a4z4 + 30a2z4 + 11z4a−2 + 28z4−2a7z3 + 16a5z3 + 35a3z3 + 29az3 + 20z3a−1 + 8z3a−3−a8z2 + 3a6z2−2a4z2−22a2z2−8z2a−2−24z2−6a5z−16a3z−17az−11za−1−4za−3 + 3a4 + 10a2 + 4a−2 + 12

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

Same Alexander/Conway Polynomial: {K11a108, K11a139, K11a231,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a231,}

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["K11a57"];
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 + 2q3−5q2 + 10q−12 + 16q−1−16q−2 + 14q−3−12q−4 + 7q−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]= {K11a108, 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]= {K11a231,}

[edit] Vassiliev invariants

V2 and V3: (1, 3)

[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 K11a57. 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          1 1
5         41 -3
3        61  5
1       64   -2
-1      106    4
-3     77     0
-5    79      -2
-7   57       2
-9  27        -5
-11 15         4
-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}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = −2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = −1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r = 1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 3 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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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K11a56

K11a58

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