K11a72

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K11a71

K11a73

Contents

Image:K11a72.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a72's page at Knotilus!

Visit K11a72's page at the original Knot Atlas!

[edit K11a72 Quick Notes]


[edit K11a72 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X12,5,13,6 X14,8,15,7 X2,10,3,9 X22,11,1,12 X18,14,19,13 X20,15,21,16 X8,18,9,17 X6,19,7,20 X16,21,17,22
Gauss code 1, -5, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, -8, 11, -6
Dowker-Thistlethwaite code 4 10 12 14 2 22 18 20 8 6 16
A Braid Representative
Image:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gif
A Morse Link Presentation Image:K11a72_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:K11a72/ThurstonBennequinNumber
Hyperbolic Volume 17.173
A-Polynomial See Data:K11a72/A-polynomial

[edit Notes for K11a72'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 K11a72's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t4−6t3 + 18t2−32t + 39−32t−1 + 18t−2−6t−3 + t−4
Conway polynomial z8 + 2z6 + 2z4 + 2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 153, 0 }
Jones polynomial q6−5q5 + 10q4−16q3 + 22q2−24q + 25−21q−1 + 15q−2−9q−3 + 4q−4−q−5
HOMFLY-PT polynomial (db, data sources) z8−a2z6−2z6a−2 + 5z6−3a2z4−6z4a−2 + z4a−4 + 10z4−3a2z2−4z2a−2 + z2a−4 + 8z2−a2 + a−2−a−4 + 2
Kauffman polynomial (db, data sources) 2z10a−2 + 2z10 + 7az9 + 14z9a−1 + 7z9a−3 + 10a2z8 + 19z8a−2 + 9z8a−4 + 20z8 + 8a3z7 + az7−16z7a−1−4z7a−3 + 5z7a−5 + 4a4z6−15a2z6−55z6a−2−20z6a−4 + z6a−6−53z6 + a5z5−12a3z5−20az5−15z5a−1−18z5a−3−10z5a−5−5a4z4 + 11a2z4 + 44z4a−2 + 12z4a−4−z4a−6 + 47z4−a5z3 + 7a3z3 + 20az3 + 22z3a−1 + 14z3a−3 + 4z3a−5 + a4z2−5a2z2−12z2a−2−2z2a−4−16z2−2a3z−5az−5za−1−za−3 + za−5 + a2−a−2−a−4 + 2
The A2 invariant −q14 + 2q12−3q10 + 2q8 + q6−4q4 + 5q2−4 + 4q−2 + q−4 + 5q−8−4q−10 + q−12−q−14−2q−16 + q−18
The G2 invariant q80−3q78 + 7q76−13q74 + 16q72−16q70 + 7q68 + 16q66−45q64 + 83q62−114q60 + 115q58−80q56−9q54 + 139q52−278q50 + 389q48−410q46 + 295q44−41q42−306q40 + 643q38−839q36 + 789q34−473q32−53q30 + 605q28−976q26 + 1019q24−679q22 + 96q20 + 491q18−837q16 + 777q14−340q12−279q10 + 797q8−952q6 + 644q4 + 28q2−796 + 1345q−2−1425q−4 + 976q−6−156q−8−748q−10 + 1412q−12−1590q−14 + 1240q−16−491q−18−358q−20 + 997q−22−1192q−24 + 906q−26−281q−28−387q−30 + 813q−32−819q−34 + 412q−36 + 228q−38−798q−40 + 1054q−42−878q−44 + 334q−46 + 339q−48−892q−50 + 1112q−52−949q−54 + 500q−56 + 44q−58−492q−60 + 703q−62−662q−64 + 443q−66−155q−68−92q−70 + 226q−72−253q−74 + 196q−76−106q−78 + 32q−80 + 21q−82−39q−84 + 35q−86−24q−88 + 11q−90−4q−92 + q−94
Further Quantum Invariants
K11a72 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["K11a72"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t4−6t3 + 18t2−32t + 39−32t−1 + 18t−2−6t−3 + t−4
In[5]:= Conway[K][z]
Out[5]= z8 + 2z6 + 2z4 + 2z2 + 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]= { 153, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q6−5q5 + 10q4−16q3 + 22q2−24q + 25−21q−1 + 15q−2−9q−3 + 4q−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 + 5z6−3a2z4−6z4a−2 + z4a−4 + 10z4−3a2z2−4z2a−2 + z2a−4 + 8z2−a2 + a−2−a−4 + 2
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 2z10a−2 + 2z10 + 7az9 + 14z9a−1 + 7z9a−3 + 10a2z8 + 19z8a−2 + 9z8a−4 + 20z8 + 8a3z7 + az7−16z7a−1−4z7a−3 + 5z7a−5 + 4a4z6−15a2z6−55z6a−2−20z6a−4 + z6a−6−53z6 + a5z5−12a3z5−20az5−15z5a−1−18z5a−3−10z5a−5−5a4z4 + 11a2z4 + 44z4a−2 + 12z4a−4−z4a−6 + 47z4−a5z3 + 7a3z3 + 20az3 + 22z3a−1 + 14z3a−3 + 4z3a−5 + a4z2−5a2z2−12z2a−2−2z2a−4−16z2−2a3z−5az−5za−1−za−3 + za−5 + a2−a−2−a−4 + 2

[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["K11a72"];
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 + 18t2−32t + 39−32t−1 + 18t−2−6t−3 + t−4, q6−5q5 + 10q4−16q3 + 22q2−24q + 25−21q−1 + 15q−2−9q−3 + 4q−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: (2, 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 K11a72. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-10123456χ
13           11
11          4 -4
9         61 5
7        104  -6
5       126   6
3      1210    -2
1     1312     1
-1    913      4
-3   612       -6
-5  39        6
-7 16         -5
-9 3          3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = −1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r = 0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
r = 1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r = 2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r = 3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r = 4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 5 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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K11a71

K11a73

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