K11a164

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K11a163

K11a165

Contents

Image:K11a164.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a164's page at Knotilus!

Visit K11a164's page at the original Knot Atlas!

[edit K11a164 Quick Notes]


[edit K11a164 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X18,6,19,5 X14,8,15,7 X16,9,17,10 X2,11,3,12 X20,13,21,14 X22,16,1,15 X12,17,13,18 X6,20,7,19 X8,21,9,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -11, 5, -2, 6, -9, 7, -4, 8, -5, 9, -3, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 10 18 14 16 2 20 22 12 6 8
A Braid Representative
Image:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart4.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gif
A Morse Link Presentation Image:K11a164_ML.gif

[edit] Three dimensional invariants

Symmetry type Chiral
Unknotting number {1,2}
3-genus 4
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a164/ThurstonBennequinNumber
Hyperbolic Volume 18.3585
A-Polynomial See Data:K11a164/A-polynomial

[edit Notes for K11a164's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [0,4]
Rasmussen s-Invariant 0

[edit Notes for K11a164's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t4−7t3 + 20t2−35t + 43−35t−1 + 20t−2−7t−3 + t−4
Conway polynomial z8 + z6−2z4−2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 169, 0 }
Jones polynomial −q5 + 5q4−11q3 + 18q2−24q + 28−27q−1 + 23q−2−17q−3 + 10q−4−4q−5 + q−6
HOMFLY-PT polynomial (db, data sources) z8−2a2z6−z6a−2 + 4z6 + a4z4−6a2z4−2z4a−2 + 5z4 + 2a4z2−5a2z2 + z2 + a4−a2 + a−2
Kauffman polynomial (db, data sources) 3a2z10 + 3z10 + 8a3z9 + 18az9 + 10z9a−1 + 8a4z8 + 16a2z8 + 14z8a−2 + 22z8 + 4a5z7−10a3z7−27az7−2z7a−1 + 11z7a−3 + a6z6−17a4z6−49a2z6−19z6a−2 + 5z6a−4−55z6−8a5z5−3a3z5 + az5−19z5a−1−14z5a−3 + z5a−5−2a6z4 + 13a4z4 + 41a2z4 + 6z4a−2−4z4a−4 + 36z4 + 5a5z3 + 5a3z3 + 8az3 + 12z3a−1 + 4z3a−3 + a6z2−6a4z2−13a2z2 + z2a−2−5z2−a5z−a3z−az−za−1 + a4 + a2−a−2
The A2 invariant q18−q16 + 3q12−4q10 + 4q8−2q6−2q4 + 4q2−5 + 6q−2−4q−4 + q−6 + 3q−8−3q−10 + 3q−12−q−14
The G2 invariant q94−3q92 + 8q90−16q88 + 23q86−28q84 + 20q82 + 10q80−63q78 + 139q76−210q74 + 232q72−166q70−19q68 + 310q66−612q64 + 808q62−756q60 + 371q58 + 267q56−968q54 + 1451q52−1462q50 + 947q48−31q46−947q44 + 1580q42−1599q40 + 975q38 + 18q36−939q34 + 1370q32−1120q30 + 316q28 + 702q26−1454q24 + 1593q22−1032q20−64q18 + 1251q16−2080q14 + 2211q12−1558q10 + 365q8 + 973q6−1971q4 + 2260q2−1767 + 683q−2 + 538q−4−1418q−6 + 1613q−8−1076q−10 + 123q−12 + 820q−14−1311q−16 + 1130q−18−397q−20−574q−22 + 1335q−24−1563q−26 + 1201q−28−394q−30−496q−32 + 1148q−34−1369q−36 + 1146q−38−627q−40 + 27q−42 + 446q−44−687q−46 + 681q−48−496q−50 + 252q−52−18q−54−141q−56 + 203q−58−199q−60 + 141q−62−73q−64 + 21q−66 + 16q−68−28q−70 + 27q−72−20q−74 + 10q−76−4q−78 + q−80
Further Quantum Invariants
K11a164 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["K11a164"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t4−7t3 + 20t2−35t + 43−35t−1 + 20t−2−7t−3 + t−4
In[5]:= Conway[K][z]
Out[5]= z8 + z6−2z4−2z2 + 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]= { 169, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q5 + 5q4−11q3 + 18q2−24q + 28−27q−1 + 23q−2−17q−3 + 10q−4−4q−5 + q−6
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z8−2a2z6−z6a−2 + 4z6 + a4z4−6a2z4−2z4a−2 + 5z4 + 2a4z2−5a2z2 + z2 + a4−a2 + a−2
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 3a2z10 + 3z10 + 8a3z9 + 18az9 + 10z9a−1 + 8a4z8 + 16a2z8 + 14z8a−2 + 22z8 + 4a5z7−10a3z7−27az7−2z7a−1 + 11z7a−3 + a6z6−17a4z6−49a2z6−19z6a−2 + 5z6a−4−55z6−8a5z5−3a3z5 + az5−19z5a−1−14z5a−3 + z5a−5−2a6z4 + 13a4z4 + 41a2z4 + 6z4a−2−4z4a−4 + 36z4 + 5a5z3 + 5a3z3 + 8az3 + 12z3a−1 + 4z3a−3 + a6z2−6a4z2−13a2z2 + z2a−2−5z2−a5z−a3z−az−za−1 + a4 + a2−a−2

[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["K11a164"];
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]= { t4−7t3 + 20t2−35t + 43−35t−1 + 20t−2−7t−3 + t−4, −q5 + 5q4−11q3 + 18q2−24q + 28−27q−1 + 23q−2−17q−3 + 10q−4−4q−5 + q−6 }
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 = 0 is the signature of K11a164. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-1012345χ
11           1-1
9          4 4
7         71 -6
5        114  7
3       137   -6
1      1511    4
-1     1314     1
-3    1014      -4
-5   713       6
-7  310        -7
-9 17         6
-11 3          -3
-131           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −1 i = 1
r = −6 {\mathbb Z}
r = −5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = −2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r = −1 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r = 0 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{15}
r = 1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r = 2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r = 3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 4 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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K11a163

K11a165

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