K11a15

From Knot Atlas

Jump to: navigation, search

K11a14

K11a16

Contents

Image:K11a15.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a15's page at Knotilus!

Visit K11a15's page at the original Knot Atlas!

[edit K11a15 Quick Notes]


[edit K11a15 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X8493 X12,5,13,6 X2837 X14,9,15,10 X18,12,19,11 X6,13,7,14 X20,15,21,16 X22,17,1,18 X10,20,11,19 X16,21,17,22
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -3, 7, -5, 8, -11, 9, -6, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 8 12 2 14 18 6 20 22 10 16
A Braid Representative
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.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:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gif
A Morse Link Presentation Image:K11a15_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:K11a15/ThurstonBennequinNumber
Hyperbolic Volume 15.1288
A-Polynomial See Data:K11a15/A-polynomial

[edit Notes for K11a15'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 K11a15's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t4 + 5t3−13t2 + 22t−25 + 22t−1−13t−2 + 5t−3−t−4
Conway polynomial −z8−3z6−3z4−z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 107, -2 }
Jones polynomial −q4 + 3q3−6q2 + 11q−14 + 17q−1−17q−2 + 15q−3−12q−4 + 7q−5−3q−6 + q−7
HOMFLY-PT polynomial (db, data sources) −a2z8 + a4z6−6a2z6 + 2z6 + 4a4z4−15a2z4−z4a−2 + 9z4 + 6a4z2−18a2z2−3z2a−2 + 14z2 + 3a4−8a2−2a−2 + 8
Kauffman polynomial (db, data sources) a2z10 + z10 + 5a3z9 + 8az9 + 3z9a−1 + 9a4z8 + 15a2z8 + 3z8a−2 + 9z8 + 9a5z7 + a3z7−15az7−6z7a−1 + z7a−3 + 6a6z6−16a4z6−52a2z6−12z6a−2−42z6 + 3a7z5−15a5z5−25a3z5−9az5−6z5a−1−4z5a−3 + a8z4−6a6z4 + 13a4z4 + 58a2z4 + 16z4a−2 + 54z4−2a7z3 + 14a5z3 + 29a3z3 + 23az3 + 15z3a−1 + 5z3a−3−a8z2 + 3a6z2−7a4z2−33a2z2−9z2a−2−31z2−5a5z−11a3z−10az−6za−1−2za−3 + 3a4 + 8a2 + 2a−2 + 8
The A2 invariant q20−q18 + 3q16−q14−q12 + q10−5q8 + 2q6−3q4 + 2q2 + 3 + 4q−4−q−6−q−12
The G2 invariant q114−2q112 + 4q110−6q108 + 6q106−5q104 + 9q100−19q98 + 29q96−37q94 + 33q92−20q90−5q88 + 44q86−78q84 + 105q82−114q80 + 87q78−31q76−56q74 + 155q72−222q70 + 240q68−180q66 + 52q64 + 114q62−253q60 + 318q58−267q56 + 116q54 + 78q52−231q50 + 278q48−189q46 + 10q44 + 180q42−295q40 + 259q38−97q36−144q34 + 352q32−446q30 + 366q28−152q26−136q24 + 378q22−501q20 + 449q18−254q16−18q14 + 262q12−389q10 + 367q8−197q6−28q4 + 220q2−291 + 223q−2−32q−4−173q−6 + 318q−8−320q−10 + 188q−12 + 29q−14−235q−16 + 359q−18−346q−20 + 222q−22−39q−24−136q−26 + 236q−28−246q−30 + 181q−32−80q−34−17q−36 + 76q−38−97q−40 + 81q−42−48q−44 + 17q−46 + 5q−48−16q−50 + 14q−52−11q−54 + 6q−56−2q−58 + q−60
Further Quantum Invariants
K11a15 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["K11a15"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t4 + 5t3−13t2 + 22t−25 + 22t−1−13t−2 + 5t−3−t−4
In[5]:= Conway[K][z]
Out[5]= −z8−3z6−3z4−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]= {1}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 107, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q4 + 3q3−6q2 + 11q−14 + 17q−1−17q−2 + 15q−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 + 9z4 + 6a4z2−18a2z2−3z2a−2 + 14z2 + 3a4−8a2−2a−2 + 8
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= a2z10 + z10 + 5a3z9 + 8az9 + 3z9a−1 + 9a4z8 + 15a2z8 + 3z8a−2 + 9z8 + 9a5z7 + a3z7−15az7−6z7a−1 + z7a−3 + 6a6z6−16a4z6−52a2z6−12z6a−2−42z6 + 3a7z5−15a5z5−25a3z5−9az5−6z5a−1−4z5a−3 + a8z4−6a6z4 + 13a4z4 + 58a2z4 + 16z4a−2 + 54z4−2a7z3 + 14a5z3 + 29a3z3 + 23az3 + 15z3a−1 + 5z3a−3−a8z2 + 3a6z2−7a4z2−33a2z2−9z2a−2−31z2−5a5z−11a3z−10az−6za−1−2za−3 + 3a4 + 8a2 + 2a−2 + 8

[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["K11a15"];
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 + 5t3−13t2 + 22t−25 + 22t−1−13t−2 + 5t−3−t−4, −q4 + 3q3−6q2 + 11q−14 + 17q−1−17q−2 + 15q−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]= {}
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, 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 K11a15. 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          2 2
5         41 -3
3        72  5
1       74   -3
-1      107    3
-3     88     0
-5    79      -2
-7   58       3
-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}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = −1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = 0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r = 1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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).

Back to the top.

K11a14

K11a16

Retrieved from "http://katlas.org/wiki/K11a15"
Personal tools