K11a332

From Knot Atlas

Jump to: navigation, search

K11a331

K11a333

Contents

Image:K11a332.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a332's page at Knotilus!

Visit K11a332's page at the original Knot Atlas!

[edit K11a332 Quick Notes]


[edit K11a332 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X6271 X14,3,15,4 X10,6,11,5 X18,7,19,8 X2,10,3,9 X22,11,1,12 X20,14,21,13 X4,15,5,16 X12,18,13,17 X8,19,9,20 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:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart4.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:K11a332_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:K11a332/ThurstonBennequinNumber
Hyperbolic Volume 19.712
A-Polynomial See Data:K11a332/A-polynomial

[edit Notes for K11a332's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [2,4]
Rasmussen s-Invariant 0

[edit Notes for K11a332's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t4−7t3 + 22t2−40t + 49−40t−1 + 22t−2−7t−3 + t−4
Conway polynomial z8 + z6 + z2 + 1
2nd Alexander ideal (db, data sources) \left\{t^2-t+1\right\}
Determinant and Signature { 189, 0 }
Jones polynomial q6−4q5 + 10q4−19q3 + 26q2−30q + 32−27q−1 + 21q−2−13q−3 + 5q−4−q−5
HOMFLY-PT polynomial (db, data sources) z8−a2z6−2z6a−2 + 4z6−2a2z4−6z4a−2 + z4a−4 + 7z4−2a2z2−7z2a−2 + 2z2a−4 + 8z2−2a2−4a−2 + a−4 + 6
Kauffman polynomial (db, data sources) 4z10a−2 + 4z10 + 14az9 + 23z9a−1 + 9z9a−3 + 19a2z8 + 17z8a−2 + 8z8a−4 + 28z8 + 13a3z7−8az7−35z7a−1−10z7a−3 + 4z7a−5 + 5a4z6−30a2z6−53z6a−2−15z6a−4 + z6a−6−72z6 + a5z5−15a3z5−18az5−6z5a−3−8z5a−5−2a4z4 + 17a2z4 + 44z4a−2 + 11z4a−4−2z4a−6 + 50z4 + 6a3z3 + 14az3 + 15z3a−1 + 13z3a−3 + 6z3a−5−6a2z2−16z2a−2−4z2a−4 + z2a−6−17z2−2a3z−5az−7za−1−6za−3−2za−5 + 2a2 + 4a−2 + a−4 + 6
The A2 invariant −q14 + 3q12−5q10 + 2q8−4q4 + 9q2−3 + 7q−2−2q−4−3q−6 + 3q−8−6q−10 + 3q−12−q−16 + q−18
The G2 invariant Data:K11a332/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11a332 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["K11a332"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t4−7t3 + 22t2−40t + 49−40t−1 + 22t−2−7t−3 + t−4
In[5]:= Conway[K][z]
Out[5]= z8 + z6 + z2 + 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\{t^2-t+1\right\}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 189, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q6−4q5 + 10q4−19q3 + 26q2−30q + 32−27q−1 + 21q−2−13q−3 + 5q−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 + 4z6−2a2z4−6z4a−2 + z4a−4 + 7z4−2a2z2−7z2a−2 + 2z2a−4 + 8z2−2a2−4a−2 + a−4 + 6
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 4z10a−2 + 4z10 + 14az9 + 23z9a−1 + 9z9a−3 + 19a2z8 + 17z8a−2 + 8z8a−4 + 28z8 + 13a3z7−8az7−35z7a−1−10z7a−3 + 4z7a−5 + 5a4z6−30a2z6−53z6a−2−15z6a−4 + z6a−6−72z6 + a5z5−15a3z5−18az5−6z5a−3−8z5a−5−2a4z4 + 17a2z4 + 44z4a−2 + 11z4a−4−2z4a−6 + 50z4 + 6a3z3 + 14az3 + 15z3a−1 + 13z3a−3 + 6z3a−5−6a2z2−16z2a−2−4z2a−4 + z2a−6−17z2−2a3z−5az−7za−1−6za−3−2za−5 + 2a2 + 4a−2 + a−4 + 6

[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["K11a332"];
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 + 22t2−40t + 49−40t−1 + 22t−2−7t−3 + t−4, q6−4q5 + 10q4−19q3 + 26q2−30q + 32−27q−1 + 21q−2−13q−3 + 5q−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]= {}
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: (1, -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 K11a332. 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         71 6
7        123  -9
5       147   7
3      1612    -4
1     1614     2
-1    1217      5
-3   915       -6
-5  412        8
-7 19         -8
-9 4          4
-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}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = −2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r = −1 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r = 0 {\mathbb Z}^{17}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{16}
r = 1 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{16} {\mathbb Z}^{16}
r = 2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r = 3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r = 4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
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).

Back to the top.

K11a331

K11a333

Retrieved from "http://katlas.org/wiki/K11a332"
Personal tools