K11a160

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K11a159

K11a161

Contents

Image:K11a160.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a160's page at Knotilus!

Visit K11a160's page at the original Knot Atlas!

[edit K11a160 Quick Notes]


[edit K11a160 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 X22,11,1,12 X20,14,21,13 X8,15,9,16 X12,18,13,17 X6,19,7,20 X16,22,17,21
Gauss code 1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -9, 7, -4, 8, -11, 9, -3, 10, -7, 11, -6
Dowker-Thistlethwaite code 4 10 18 14 2 22 20 8 12 6 16
A Braid Representative
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gif
A Morse Link Presentation Image:K11a160_ML.gif

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 4
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a160/ThurstonBennequinNumber
Hyperbolic Volume 17.0985
A-Polynomial See Data:K11a160/A-polynomial

[edit Notes for K11a160's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 0

[edit Notes for K11a160's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t4−6t3 + 17t2−30t + 37−30t−1 + 17t−2−6t−3 + t−4
Conway polynomial z8 + 2z6 + z4 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 145, 0 }
Jones polynomial q6−4q5 + 9q4−15q3 + 20q2−23q + 24−20q−1 + 15q−2−9q−3 + 4q−4−q−5
HOMFLY-PT polynomial (db, data sources) z8−a2z6−2z6a−2 + 5z6−3a2z4−7z4a−2 + z4a−4 + 10z4−3a2z2−8z2a−2 + 2z2a−4 + 9z2−a2−3a−2 + a−4 + 4
Kauffman polynomial (db, data sources) 2z10a−2 + 2z10 + 7az9 + 13z9a−1 + 6z9a−3 + 10a2z8 + 15z8a−2 + 7z8a−4 + 18z8 + 8a3z7−az7−17z7a−1−4z7a−3 + 4z7a−5 + 4a4z6−16a2z6−47z6a−2−15z6a−4 + z6a−6−51z6 + a5z5−12a3z5−17az5−11z5a−1−16z5a−3−9z5a−5−5a4z4 + 12a2z4 + 44z4a−2 + 9z4a−4−2z4a−6 + 50z4−a5z3 + 6a3z3 + 19az3 + 24z3a−1 + 18z3a−3 + 6z3a−5 + a4z2−6a2z2−18z2a−2−4z2a−4 + z2a−6−20z2−2a3z−6az−8za−1−6za−3−2za−5 + a2 + 3a−2 + a−4 + 4
The A2 invariant −q14 + 2q12−3q10 + 2q8 + q6−3q4 + 6q2−3 + 4q−2−q−4−2q−6 + 3q−8−4q−10 + 2q−12−q−16 + q−18
The G2 invariant q80−3q78 + 7q76−13q74 + 16q72−16q70 + 7q68 + 16q66−46q64 + 84q62−113q60 + 111q58−73q56−15q54 + 143q52−275q50 + 376q48−386q46 + 258q44−5q42−327q40 + 628q38−780q36 + 692q34−355q32−146q30 + 635q28−907q26 + 852q24−467q22−87q20 + 565q18−770q16 + 591q14−104q12−456q10 + 848q8−849q6 + 439q4 + 234q2−898 + 1280q−2−1218q−4 + 713q−6 + 58q−8−812q−10 + 1298q−12−1328q−14 + 916q−16−225q−18−485q−20 + 928q−22−967q−24 + 608q−26−22q−28−518q−30 + 780q−32−647q−34 + 179q−36 + 404q−38−847q−40 + 945q−42−667q−44 + 124q−46 + 452q−48−845q−50 + 935q−52−705q−54 + 288q−56 + 144q−58−455q−60 + 556q−62−473q−64 + 285q−66−69q−68−91q−70 + 170q−72−173q−74 + 127q−76−66q−78 + 18q−80 + 13q−82−25q−84 + 22q−86−16q−88 + 8q−90−3q−92 + q−94
Further Quantum Invariants
K11a160 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["K11a160"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t4−6t3 + 17t2−30t + 37−30t−1 + 17t−2−6t−3 + t−4
In[5]:= Conway[K][z]
Out[5]= z8 + 2z6 + z4 + 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]= { 145, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q6−4q5 + 9q4−15q3 + 20q2−23q + 24−20q−1 + 15q−2−9q−3 + 4q−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 + 5z6−3a2z4−7z4a−2 + z4a−4 + 10z4−3a2z2−8z2a−2 + 2z2a−4 + 9z2−a2−3a−2 + a−4 + 4
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 2z10a−2 + 2z10 + 7az9 + 13z9a−1 + 6z9a−3 + 10a2z8 + 15z8a−2 + 7z8a−4 + 18z8 + 8a3z7−az7−17z7a−1−4z7a−3 + 4z7a−5 + 4a4z6−16a2z6−47z6a−2−15z6a−4 + z6a−6−51z6 + a5z5−12a3z5−17az5−11z5a−1−16z5a−3−9z5a−5−5a4z4 + 12a2z4 + 44z4a−2 + 9z4a−4−2z4a−6 + 50z4−a5z3 + 6a3z3 + 19az3 + 24z3a−1 + 18z3a−3 + 6z3a−5 + a4z2−6a2z2−18z2a−2−4z2a−4 + z2a−6−20z2−2a3z−6az−8za−1−6za−3−2za−5 + a2 + 3a−2 + a−4 + 4

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

Same Alexander/Conway Polynomial: {K11a76, K11a289,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a76, K11a289,}

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["K11a160"];
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]= { t4−6t3 + 17t2−30t + 37−30t−1 + 17t−2−6t−3 + t−4, q6−4q5 + 9q4−15q3 + 20q2−23q + 24−20q−1 + 15q−2−9q−3 + 4q−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]= {K11a76, K11a289,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11a76, K11a289,}

[edit] Vassiliev invariants

V2 and V3: (0, -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 = 0 is the signature of K11a160. 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         61 5
7        93  -6
5       116   5
3      129    -3
1     1211     1
-1    913      4
-3   611       -5
-5  39        6
-7 16         -5
-9 3          3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = −1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r = 0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r = 1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r = 2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r = 3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r = 4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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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K11a159

K11a161

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