K11a47

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K11a46

K11a48

Contents

Image:K11a47.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a47's page at Knotilus!

Visit K11a47's page at the original Knot Atlas!

[edit K11a47 Quick Notes]


[edit K11a47 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X8493 X14,5,15,6 X2837 X20,9,21,10 X22,11,1,12 X18,14,19,13 X6,15,7,16 X12,18,13,17 X16,20,17,19 X10,21,11,22
Gauss code 1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -9, 7, -3, 8, -10, 9, -7, 10, -5, 11, -6
Dowker-Thistlethwaite code 4 8 14 2 20 22 18 6 12 16 10
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:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gif
A Morse Link Presentation Image:K11a47_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:K11a47/ThurstonBennequinNumber
Hyperbolic Volume 15.5977
A-Polynomial See Data:K11a47/A-polynomial

[edit Notes for K11a47'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 K11a47's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t4−5t3 + 14t2−24t + 29−24t−1 + 14t−2−5t−3 + t−4
Conway polynomial z8 + 3z6 + 4z4 + 3z2 + 1
2nd Alexander ideal (db, data sources) \left\{t^2-t+1\right\}
Determinant and Signature { 117, 0 }
Jones polynomial q6−3q5 + 6q4−12q3 + 16q2−18q + 20−16q−1 + 13q−2−8q−3 + 3q−4−q−5
HOMFLY-PT polynomial (db, data sources) z8−a2z6−2z6a−2 + 6z6−4a2z4−9z4a−2 + z4a−4 + 16z4−7a2z2−15z2a−2 + 3z2a−4 + 22z2−5a2−9a−2 + 2a−4 + 13
Kauffman polynomial (db, data sources) z10a−2 + z10 + 4az9 + 7z9a−1 + 3z9a−3 + 7a2z8 + 10z8a−2 + 4z8a−4 + 13z8 + 6a3z7 + 5az7−2z7a−1 + 2z7a−3 + 3z7a−5 + 3a4z6−11a2z6−28z6a−2−8z6a−4 + z6a−6−33z6 + a5z5−10a3z5−25az5−27z5a−1−22z5a−3−9z5a−5−4a4z4 + 11a2z4 + 28z4a−2 + 3z4a−4−3z4a−6 + 37z4−2a5z3 + 8a3z3 + 31az3 + 42z3a−1 + 30z3a−3 + 9z3a−5 + a4z2−11a2z2−19z2a−2−z2a−4 + 2z2a−6−28z2 + a5z−4a3z−15az−21za−1−15za−3−4za−5 + 5a2 + 9a−2 + 2a−4 + 13
The A2 invariant −q14 + q12−4q10−q4 + 8q2 + 1 + 6q−2−q−4−3q−6−5q−10 + q−12 + q−18
The G2 invariant q80−2q78 + 5q76−8q74 + 10q72−10q70 + 4q68 + 11q66−32q64 + 56q62−76q60 + 72q58−44q56−17q54 + 110q52−199q50 + 259q48−246q46 + 131q44 + 52q42−273q40 + 435q38−479q36 + 360q34−112q32−197q30 + 437q28−512q26 + 392q24−123q22−181q20 + 367q18−368q16 + 183q14 + 119q12−385q10 + 511q8−392q6 + 97q4 + 299q2−617 + 749q−2−608q−4 + 270q−6 + 167q−8−537q−10 + 735q−12−661q−14 + 376q−16 + 9q−18−349q−20 + 498q−22−427q−24 + 162q−26 + 140q−28−358q−30 + 388q−32−232q−34−64q−36 + 347q−38−505q−40 + 463q−42−264q−44−42q−46 + 310q−48−453q−50 + 451q−52−307q−54 + 106q−56 + 93q−58−223q−60 + 258q−62−217q−64 + 131q−66−34q−68−36q−70 + 74q−72−78q−74 + 63q−76−35q−78 + 13q−80 + 3q−82−12q−84 + 10q−86−9q−88 + 5q−90−2q−92 + q−94
Further Quantum Invariants
K11a47 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["K11a47"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t4−5t3 + 14t2−24t + 29−24t−1 + 14t−2−5t−3 + t−4
In[5]:= Conway[K][z]
Out[5]= z8 + 3z6 + 4z4 + 3z2 + 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]= { 117, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q6−3q5 + 6q4−12q3 + 16q2−18q + 20−16q−1 + 13q−2−8q−3 + 3q−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 + 6z6−4a2z4−9z4a−2 + z4a−4 + 16z4−7a2z2−15z2a−2 + 3z2a−4 + 22z2−5a2−9a−2 + 2a−4 + 13
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z10a−2 + z10 + 4az9 + 7z9a−1 + 3z9a−3 + 7a2z8 + 10z8a−2 + 4z8a−4 + 13z8 + 6a3z7 + 5az7−2z7a−1 + 2z7a−3 + 3z7a−5 + 3a4z6−11a2z6−28z6a−2−8z6a−4 + z6a−6−33z6 + a5z5−10a3z5−25az5−27z5a−1−22z5a−3−9z5a−5−4a4z4 + 11a2z4 + 28z4a−2 + 3z4a−4−3z4a−6 + 37z4−2a5z3 + 8a3z3 + 31az3 + 42z3a−1 + 30z3a−3 + 9z3a−5 + a4z2−11a2z2−19z2a−2−z2a−4 + 2z2a−6−28z2 + a5z−4a3z−15az−21za−1−15za−3−4za−5 + 5a2 + 9a−2 + 2a−4 + 13

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

Same Alexander/Conway Polynomial: {K11a44, K11a109,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a44,}

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["K11a47"];
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 + 14t2−24t + 29−24t−1 + 14t−2−5t−3 + t−4, q6−3q5 + 6q4−12q3 + 16q2−18q + 20−16q−1 + 13q−2−8q−3 + 3q−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]= {K11a44, K11a109,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11a44,}

[edit] Vassiliev invariants

V2 and V3: (3, -2)

[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 K11a47. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-10123456χ
13           11
11          2 -2
9         41 3
7        82  -6
5       84   4
3      108    -2
1     108     2
-1    711      4
-3   69       -3
-5  27        5
-7 16         -5
-9 2          2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = −1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r = 1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r = 2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = 3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = 4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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).

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K11a46

K11a48

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