K11a55

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K11a54

K11a56

Contents

Image:K11a55.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a55's page at Knotilus!

Visit K11a55's page at the original Knot Atlas!

[edit K11a55 Quick Notes]


[edit K11a55 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X8493 X16,6,17,5 X2837 X18,9,19,10 X20,11,21,12 X22,13,1,14 X6,16,7,15 X14,17,15,18 X10,19,11,20 X12,21,13,22
Gauss code 1, -4, 2, -1, 3, -8, 4, -2, 5, -10, 6, -11, 7, -9, 8, -3, 9, -5, 10, -6, 11, -7
Dowker-Thistlethwaite code 4 8 16 2 18 20 22 6 14 10 12
A Braid Representative
Image:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart2.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart0.gif
A Morse Link Presentation Image:K11a55_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:K11a55/ThurstonBennequinNumber
Hyperbolic Volume 11.6846
A-Polynomial See Data:K11a55/A-polynomial

[edit Notes for K11a55'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 K11a55's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t4 + 5t3−10t2 + 13t−13 + 13t−1−10t−2 + 5t−3−t−4
Conway polynomial −z8−3z6 + 2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 71, -2 }
Jones polynomial −q4 + 2q3−4q2 + 7q−8 + 11q−1−11q−2 + 10q−3−8q−4 + 5q−5−3q−6 + q−7
HOMFLY-PT polynomial (db, data sources) −a2z8 + a4z6−6a2z6 + 2z6 + 4a4z4−13a2z4−z4a−2 + 10z4 + 4a4z2−13a2z2−4z2a−2 + 15z2 + a4−5a2−3a−2 + 8
Kauffman polynomial (db, data sources) a2z10 + z10 + 3a3z9 + 5az9 + 2z9a−1 + 4a4z8 + 3a2z8 + 2z8a−2 + z8 + 4a5z7−6a3z7−17az7−6z7a−1 + z7a−3 + 4a6z6−7a4z6−19a2z6−9z6a−2−17z6 + 3a7z5−3a5z5 + 4a3z5 + 14az5−z5a−1−5z5a−3 + a8z4−4a6z4 + 7a4z4 + 29a2z4 + 12z4a−2 + 29z4−4a7z3−2a5z3 + a3z3 + 2az3 + 10z3a−1 + 7z3a−3−a8z2−4a4z2−19a2z2−8z2a−2−22z2 + a7z + 2a5z−a3z−5az−6za−1−3za−3 + a4 + 5a2 + 3a−2 + 8
The A2 invariant q20−q18 + q16−q14−q12−3q8 + 2q6−q4 + 3q2 + 3 + q−2 + 2q−4−q−6−q−10−q−12
The G2 invariant q114−2q112 + 4q110−6q108 + 4q106−2q104−4q102 + 12q100−18q98 + 22q96−19q94 + 8q92 + 7q90−23q88 + 37q86−40q84 + 34q82−19q80−2q78 + 23q76−36q74 + 46q72−44q70 + 33q68−16q66−8q64 + 30q62−44q60 + 48q58−35q56 + 11q54 + 15q52−37q50 + 35q48−16q46−16q44 + 41q42−49q40 + 26q38 + 11q36−55q34 + 81q32−86q30 + 51q28−2q26−53q24 + 93q22−101q20 + 80q18−33q16−17q14 + 59q12−77q10 + 73q8−34q6−7q4 + 46q2−52 + 39q−2 + 6q−4−44q−6 + 68q−8−59q−10 + 26q−12 + 25q−14−68q−16 + 92q−18−78q−20 + 42q−22 + 3q−24−47q−26 + 66q−28−65q−30 + 45q−32−19q−34−7q−36 + 22q−38−29q−40 + 23q−42−16q−44 + 6q−46−5q−50 + 4q−52−4q−54 + 3q−56−q−58 + q−60
Further Quantum Invariants
K11a55 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["K11a55"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t4 + 5t3−10t2 + 13t−13 + 13t−1−10t−2 + 5t−3−t−4
In[5]:= Conway[K][z]
Out[5]= −z8−3z6 + 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]= { 71, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q4 + 2q3−4q2 + 7q−8 + 11q−1−11q−2 + 10q−3−8q−4 + 5q−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 + 4a4z4−13a2z4−z4a−2 + 10z4 + 4a4z2−13a2z2−4z2a−2 + 15z2 + a4−5a2−3a−2 + 8
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= a2z10 + z10 + 3a3z9 + 5az9 + 2z9a−1 + 4a4z8 + 3a2z8 + 2z8a−2 + z8 + 4a5z7−6a3z7−17az7−6z7a−1 + z7a−3 + 4a6z6−7a4z6−19a2z6−9z6a−2−17z6 + 3a7z5−3a5z5 + 4a3z5 + 14az5−z5a−1−5z5a−3 + a8z4−4a6z4 + 7a4z4 + 29a2z4 + 12z4a−2 + 29z4−4a7z3−2a5z3 + a3z3 + 2az3 + 10z3a−1 + 7z3a−3−a8z2−4a4z2−19a2z2−8z2a−2−22z2 + a7z + 2a5z−a3z−5az−6za−1−3za−3 + a4 + 5a2 + 3a−2 + 8

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

Same Alexander/Conway Polynomial: {}

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["K11a55"];
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 + 5t3−10t2 + 13t−13 + 13t−1−10t−2 + 5t−3−t−4, −q4 + 2q3−4q2 + 7q−8 + 11q−1−11q−2 + 10q−3−8q−4 + 5q−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]= {}
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: (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 = -2 is the signature of K11a55. 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         31 -2
3        41  3
1       43   -1
-1      74    3
-3     55     0
-5    56      -1
-7   35       2
-9  25        -3
-11 13         2
-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}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = −1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r = 1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11a54

K11a56

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