K11a158

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K11a157

K11a159

Contents

Image:K11a158.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a158's page at Knotilus!

Visit K11a158's page at the original Knot Atlas!

[edit K11a158 Quick Notes]


[edit K11a158 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X18,5,19,6 X14,7,15,8 X2,10,3,9 X16,11,17,12 X20,14,21,13 X8,15,9,16 X22,17,1,18 X6,19,7,20 X12,22,13,21
Gauss code 1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -11, 7, -4, 8, -6, 9, -3, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 10 18 14 2 16 20 8 22 6 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:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart4.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gif
A Morse Link Presentation Image:K11a158_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:K11a158/ThurstonBennequinNumber
Hyperbolic Volume 16.235
A-Polynomial See Data:K11a158/A-polynomial

[edit Notes for K11a158'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 K11a158's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t4 + 5t3−14t2 + 25t−29 + 25t−1−14t−2 + 5t−3−t−4
Conway polynomial −z8−3z6−4z4−2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 119, -2 }
Jones polynomial −q4 + 4q3−8q2 + 13q−16 + 19q−1−19q−2 + 16q−3−12q−4 + 7q−5−3q−6 + q−7
HOMFLY-PT polynomial (db, data sources) −a2z8 + a4z6−6a2z6 + 2z6 + 4a4z4−15a2z4−z4a−2 + 8z4 + 6a4z2−17a2z2−2z2a−2 + 11z2 + 3a4−7a2−a−2 + 6
Kauffman polynomial (db, data sources) 2a2z10 + 2z10 + 7a3z9 + 12az9 + 5z9a−1 + 10a4z8 + 14a2z8 + 4z8a−2 + 8z8 + 9a5z7−7a3z7−30az7−13z7a−1 + z7a−3 + 6a6z6−19a4z6−57a2z6−14z6a−2−46z6 + 3a7z5−14a5z5−12a3z5 + 10az5 + 2z5a−1−3z5a−3 + a8z4−6a6z4 + 17a4z4 + 65a2z4 + 15z4a−2 + 56z4−2a7z3 + 12a5z3 + 18a3z3 + 10az3 + 9z3a−1 + 3z3a−3−a8z2 + 3a6z2−9a4z2−34a2z2−6z2a−2−27z2−4a5z−7a3z−5az−3za−1−za−3 + 3a4 + 7a2 + a−2 + 6
The A2 invariant q20−q18 + 3q16−q14−q12 + 2q10−5q8 + 2q6−3q4 + q2 + 3−q−2 + 4q−4−q−6 + q−10−q−12
The G2 invariant q114−2q112 + 4q110−6q108 + 6q106−5q104 + 9q100−19q98 + 29q96−36q94 + 31q92−18q90−6q88 + 41q86−71q84 + 95q82−101q80 + 80q78−38q76−28q74 + 110q72−184q70 + 232q68−221q66 + 138q64 + 15q62−189q60 + 341q58−389q56 + 299q54−92q52−167q50 + 361q48−401q46 + 268q44−5q42−260q40 + 396q38−338q36 + 88q34 + 233q32−489q30 + 543q28−376q26 + 39q24 + 334q22−601q20 + 659q18−500q16 + 167q14 + 201q12−487q10 + 590q8−473q6 + 202q4 + 126q2−368 + 438q−2−298q−4 + 21q−6 + 276q−8−444q−10 + 409q−12−173q−14−149q−16 + 430q−18−533q−20 + 435q−22−184q−24−118q−26 + 340q−28−418q−30 + 346q−32−175q−34−7q−36 + 131q−38−178q−40 + 153q−42−91q−44 + 30q−46 + 14q−48−32q−50 + 28q−52−19q−54 + 9q−56−3q−58 + q−60
Further Quantum Invariants
K11a158 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["K11a158"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t4 + 5t3−14t2 + 25t−29 + 25t−1−14t−2 + 5t−3−t−4
In[5]:= Conway[K][z]
Out[5]= −z8−3z6−4z4−2z2 + 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]= { 119, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q4 + 4q3−8q2 + 13q−16 + 19q−1−19q−2 + 16q−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 + 8z4 + 6a4z2−17a2z2−2z2a−2 + 11z2 + 3a4−7a2−a−2 + 6
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 2a2z10 + 2z10 + 7a3z9 + 12az9 + 5z9a−1 + 10a4z8 + 14a2z8 + 4z8a−2 + 8z8 + 9a5z7−7a3z7−30az7−13z7a−1 + z7a−3 + 6a6z6−19a4z6−57a2z6−14z6a−2−46z6 + 3a7z5−14a5z5−12a3z5 + 10az5 + 2z5a−1−3z5a−3 + a8z4−6a6z4 + 17a4z4 + 65a2z4 + 15z4a−2 + 56z4−2a7z3 + 12a5z3 + 18a3z3 + 10az3 + 9z3a−1 + 3z3a−3−a8z2 + 3a6z2−9a4z2−34a2z2−6z2a−2−27z2−4a5z−7a3z−5az−3za−1−za−3 + 3a4 + 7a2 + a−2 + 6

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

Same Alexander/Conway Polynomial: {K11a34,}

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["K11a158"];
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 + 25t−29 + 25t−1−14t−2 + 5t−3−t−4, −q4 + 4q3−8q2 + 13q−16 + 19q−1−19q−2 + 16q−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]= {K11a34,}
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, 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 K11a158. 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          3 3
5         51 -4
3        83  5
1       85   -3
-1      118    3
-3     99     0
-5    710      -3
-7   59       4
-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}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = −1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r = 0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r = 1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = 2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
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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K11a157

K11a159

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