K11a125

From Knot Atlas

Jump to: navigation, search

K11a124

K11a126

Contents

Image:K11a125.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a125's page at Knotilus!

Visit K11a125's page at the original Knot Atlas!

[edit K11a125 Quick Notes]


[edit K11a125 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,5,15,6 X20,8,21,7 X2,10,3,9 X16,11,17,12 X18,14,19,13 X8,15,9,16 X22,17,1,18 X6,20,7,19 X12,22,13,21
Gauss code 1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -11, 7, -3, 8, -6, 9, -7, 10, -4, 11, -9
Dowker-Thistlethwaite code 4 10 14 20 2 16 18 8 22 6 12
A Braid Representative
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart4.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart0.gif
A Morse Link Presentation Image:K11a125_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:K11a125/ThurstonBennequinNumber
Hyperbolic Volume 18.6455
A-Polynomial See Data:K11a125/A-polynomial

[edit Notes for K11a125'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 K11a125's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial −t4 + 6t3−19t2 + 38t−47 + 38t−1−19t−2 + 6t−3−t−4
Conway polynomial −z8−2z6−3z4 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 175, 2 }
Jones polynomial −q8 + 5q7−12q6 + 19q5−25q4 + 29q3−28q2 + 24q−17 + 10q−1−4q−2 + q−3
HOMFLY-PT polynomial (db, data sources) −z8a−2−5z6a−2 + 2z6a−4 + z6−11z4a−2 + 6z4a−4−z4a−6 + 3z4−10z2a−2 + 7z2a−4−z2a−6 + 4z2−3a−2 + 3a−4−a−6 + 2
Kauffman polynomial (db, data sources) 3z10a−2 + 3z10a−4 + 8z9a−1 + 19z9a−3 + 11z9a−5 + 18z8a−2 + 26z8a−4 + 16z8a−6 + 8z8 + 4az7−9z7a−1−27z7a−3−2z7a−5 + 12z7a−7 + a2z6−55z6a−2−66z6a−4−24z6a−6 + 5z6a−8−17z6−8az5−5z5a−1−3z5a−3−22z5a−5−15z5a−7 + z5a−9−2a2z4 + 50z4a−2 + 49z4a−4 + 12z4a−6−3z4a−8 + 14z4 + 5az3 + 8z3a−1 + 12z3a−3 + 14z3a−5 + 5z3a−7 + a2z2−20z2a−2−16z2a−4−4z2a−6−7z2−az−2za−1−2za−3−2za−5−za−7 + 3a−2 + 3a−4 + a−6 + 2
The A2 invariant q8−2q6 + 4q4−2q2−1 + 5q−2−6q−4 + 5q−6−3q−8 + q−10 + 3q−12−4q−14 + 5q−16−3q−18−q−20 + 2q−22−q−24
The G2 invariant q46−3q44 + 8q42−16q40 + 23q38−28q36 + 20q34 + 10q32−62q30 + 137q28−208q26 + 233q24−176q22−2q20 + 294q18−608q16 + 833q14−812q12 + 451q10 + 199q8−958q6 + 1532q4−1632q2 + 1151−188q−2−916q−4 + 1717q−6−1870q−8 + 1282q−10−198q−12−912q−14 + 1540q−16−1415q−18 + 595q−20 + 565q−22−1513q−24 + 1825q−26−1319q−28 + 142q−30 + 1226q−32−2263q−34 + 2534q−36−1901q−38 + 590q−40 + 955q−42−2166q−44 + 2616q−46−2157q−48 + 971q−50 + 450q−52−1558q−54 + 1919q−56−1425q−58 + 370q−60 + 777q−62−1465q−64 + 1405q−66−655q−68−462q−70 + 1422q−72−1815q−74 + 1496q−76−606q−78−462q−80 + 1302q−82−1636q−84 + 1413q−86−801q−88 + 61q−90 + 535q−92−847q−94 + 837q−96−597q−98 + 284q−100 + 7q−102−189q−104 + 249q−106−230q−108 + 153q−110−72q−112 + 14q−114 + 22q−116−31q−118 + 28q−120−20q−122 + 10q−124−4q−126 + q−128
Further Quantum Invariants
K11a125 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["K11a125"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= −t4 + 6t3−19t2 + 38t−47 + 38t−1−19t−2 + 6t−3−t−4
In[5]:= Conway[K][z]
Out[5]= −z8−2z6−3z4 + 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]= { 175, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q8 + 5q7−12q6 + 19q5−25q4 + 29q3−28q2 + 24q−17 + 10q−1−4q−2 + q−3
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= −z8a−2−5z6a−2 + 2z6a−4 + z6−11z4a−2 + 6z4a−4−z4a−6 + 3z4−10z2a−2 + 7z2a−4−z2a−6 + 4z2−3a−2 + 3a−4−a−6 + 2
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 3z10a−2 + 3z10a−4 + 8z9a−1 + 19z9a−3 + 11z9a−5 + 18z8a−2 + 26z8a−4 + 16z8a−6 + 8z8 + 4az7−9z7a−1−27z7a−3−2z7a−5 + 12z7a−7 + a2z6−55z6a−2−66z6a−4−24z6a−6 + 5z6a−8−17z6−8az5−5z5a−1−3z5a−3−22z5a−5−15z5a−7 + z5a−9−2a2z4 + 50z4a−2 + 49z4a−4 + 12z4a−6−3z4a−8 + 14z4 + 5az3 + 8z3a−1 + 12z3a−3 + 14z3a−5 + 5z3a−7 + a2z2−20z2a−2−16z2a−4−4z2a−6−7z2−az−2za−1−2za−3−2za−5−za−7 + 3a−2 + 3a−4 + a−6 + 2

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a297,}

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["K11a125"];
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 + 6t3−19t2 + 38t−47 + 38t−1−19t−2 + 6t−3−t−4, −q8 + 5q7−12q6 + 19q5−25q4 + 29q3−28q2 + 24q−17 + 10q−1−4q−2 + q−3 }
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]= {K11a297,}

[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 = 2 is the signature of K11a125. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-101234567χ
17           1-1
15          4 4
13         81 -7
11        114  7
9       148   -6
7      1511    4
5     1314     1
3    1115      -4
1   714       7
-1  310        -7
-3 17         6
-5 3          -3
-71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 1 i = 3
r = −4 {\mathbb Z}
r = −3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 0 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r = 1 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r = 2 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r = 3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r = 4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r = 5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = 6 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 7 {\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.

K11a124

K11a126

Retrieved from "http://katlas.org/wiki/K11a125"
Personal tools