K11n183

From Knot Atlas

Jump to: navigation, search

K11n182

K11n184

Contents

Image:K11n183.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n183's page at Knotilus!

Visit K11n183's page at the original Knot Atlas!

[edit K11n183 Quick Notes]


[edit K11n183 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X6271 X3,15,4,14 X10,6,11,5 X7,19,8,18 X2,10,3,9 X22,11,1,12 X20,14,21,13 X15,5,16,4 X12,18,13,17 X19,9,20,8 X16,22,17,21
Gauss code 1, -5, -2, 8, 3, -1, -4, 10, 5, -3, 6, -9, 7, 2, -8, -11, 9, 4, -10, -7, 11, -6
Dowker-Thistlethwaite code 6 -14 10 -18 2 22 20 -4 12 -8 16
A Braid Representative
Image:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gif
A Morse Link Presentation Image:K11n183_ML.gif

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n183/ThurstonBennequinNumber
Hyperbolic Volume 11.4109
A-Polynomial See Data:K11n183/A-polynomial

[edit Notes for K11n183's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -6

[edit Notes for K11n183's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t3 + t2−6t + 9−6t−1 + t−2 + t−3
Conway polynomial z6 + 7z4 + 7z2 + 1
2nd Alexander ideal (db, data sources) \left\{4,t^2+t+1\right\}
Determinant and Signature { 21, 4 }
Jones polynomial q12−3q11 + 3q10−4q9 + 4q8−3q7 + 3q6−q5 + q3
HOMFLY-PT polynomial (db, data sources) z6a−6 + 6z4a−6 + z4a−8 + 8z2a−6 + 2z2a−8−3z2a−10 + 2a−6 + 2a−8−4a−10 + a−12
Kauffman polynomial (db, data sources) 2z8a−10 + 2z8a−12 + 2z7a−9 + 5z7a−11 + 3z7a−13 + z6a−6−z6a−8−10z6a−10−7z6a−12 + z6a−14−z5a−7−11z5a−9−22z5a−11−12z5a−13−6z4a−6 + 4z4a−8 + 17z4a−10 + 4z4a−12−3z4a−14 + 2z3a−7 + 18z3a−9 + 26z3a−11 + 10z3a−13 + 8z2a−6−4z2a−8−14z2a−10−z2a−12 + z2a−14−8za−9−9za−11−za−13−2a−6 + 2a−8 + 4a−10 + a−12
The A2 invariant q−10 + q−12 + q−16 + q−18 + 3q−22 + q−24 + q−26−2q−28−3q−30−q−32−2q−34 + q−36 + q−38
The G2 invariant Data:K11n183/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n183 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["K11n183"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t3 + t2−6t + 9−6t−1 + t−2 + t−3
In[5]:= Conway[K][z]
Out[5]= z6 + 7z4 + 7z2 + 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\{4,t^2+t+1\right\}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 21, 4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q12−3q11 + 3q10−4q9 + 4q8−3q7 + 3q6−q5 + q3
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z6a−6 + 6z4a−6 + z4a−8 + 8z2a−6 + 2z2a−8−3z2a−10 + 2a−6 + 2a−8−4a−10 + a−12
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 2z8a−10 + 2z8a−12 + 2z7a−9 + 5z7a−11 + 3z7a−13 + z6a−6−z6a−8−10z6a−10−7z6a−12 + z6a−14−z5a−7−11z5a−9−22z5a−11−12z5a−13−6z4a−6 + 4z4a−8 + 17z4a−10 + 4z4a−12−3z4a−14 + 2z3a−7 + 18z3a−9 + 26z3a−11 + 10z3a−13 + 8z2a−6−4z2a−8−14z2a−10−z2a−12 + z2a−14−8za−9−9za−11−za−13−2a−6 + 2a−8 + 4a−10 + a−12

[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["K11n183"];
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]= { t3 + t2−6t + 9−6t−1 + t−2 + t−3, q12−3q11 + 3q10−4q9 + 4q8−3q7 + 3q6−q5 + q3 }
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: (7, 17)

[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 = 4 is the signature of K11n183. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 012345678910χ
25          11
23         2 -2
21        11 0
19       32  -1
17     121   0
15     23    1
13   132     0
11    2      2
9  12       -1
71          1
51          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 3 i = 5 i = 7
r = 0 {\mathbb Z} {\mathbb Z}
r = 1
r = 2 {\mathbb Z}
r = 3 {\mathbb Z}^{2} {\mathbb Z}_2 {\mathbb Z}
r = 4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r = 5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2} {\mathbb Z}
r = 6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = 9 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 10 {\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).

Back to the top.

K11n182

K11n184

Retrieved from "http://katlas.org/wiki/K11n183"
Personal tools