K11a73

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K11a72

K11a74

Contents

Image:K11a73.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a73's page at Knotilus!

Visit K11a73's page at the original Knot Atlas!

[edit K11a73 Quick Notes]


[edit K11a73 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X16,10,17,9 X2,11,3,12 X18,13,19,14 X20,16,21,15 X22,17,1,18 X6,20,7,19 X8,21,9,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -11, 5, -2, 6, -3, 7, -4, 8, -5, 9, -7, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 10 12 14 16 2 18 20 22 6 8
A Braid Representative
Image:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gif
A Morse Link Presentation Image:K11a73_ML.gif

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 4
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a73/ThurstonBennequinNumber
Hyperbolic Volume 18.4184
A-Polynomial See Data:K11a73/A-polynomial

[edit Notes for K11a73's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 0

[edit Notes for K11a73's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t4−7t3 + 21t2−37t + 45−37t−1 + 21t−2−7t−3 + t−4
Conway polynomial z8 + z6−z4 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 177, 0 }
Jones polynomial −q5 + 5q4−11q3 + 18q2−25q + 29−28q−1 + 25q−2−18q−3 + 11q−4−5q−5 + q−6
HOMFLY-PT polynomial (db, data sources) z8−2a2z6−z6a−2 + 4z6 + a4z4−5a2z4−2z4a−2 + 5z4 + a4z2−a2z2−a4 + 3a2 + a−2−2
Kauffman polynomial (db, data sources) 3a2z10 + 3z10 + 9a3z9 + 19az9 + 10z9a−1 + 10a4z8 + 21a2z8 + 14z8a−2 + 25z8 + 5a5z7−8a3z7−23az7 + z7a−1 + 11z7a−3 + a6z6−21a4z6−60a2z6−18z6a−2 + 5z6a−4−61z6−9a5z5−14a3z5−16az5−26z5a−1−14z5a−3 + z5a−5−a6z4 + 13a4z4 + 42a2z4 + 6z4a−2−4z4a−4 + 38z4 + 4a5z3 + 14a3z3 + 23az3 + 18z3a−1 + 5z3a−3−2a4z2−5a2z2−3z2−2a3z−4az−2za−1−a4−3a2−a−2−2
The A2 invariant q18−2q16−q14 + 2q12−4q10 + 6q8 + 4q2−6 + 5q−2−5q−4 + q−6 + 3q−8−3q−10 + 3q−12−q−14
The G2 invariant q94−4q92 + 11q90−24q88 + 36q86−43q84 + 30q82 + 20q80−101q78 + 212q76−304q74 + 312q72−190q70−98q68 + 491q66−854q64 + 1023q62−847q60 + 283q58 + 525q56−1305q54 + 1734q52−1579q50 + 834q48 + 250q46−1270q44 + 1787q42−1587q40 + 759q38 + 346q36−1218q34 + 1474q32−989q30−18q28 + 1115q26−1794q24 + 1733q22−901q20−408q18 + 1698q16−2465q14 + 2411q12−1509q10 + 86q8 + 1352q6−2298q4 + 2402q2−1665 + 400q−2 + 860q−4−1617q−6 + 1594q−8−856q−10−216q−12 + 1119q−14−1449q−16 + 1056q−18−149q−20−885q−22 + 1595q−24−1680q−26 + 1162q−28−252q−30−678q−32 + 1298q−34−1452q−36 + 1156q−38−583q−40−34q−42 + 503q−44−720q−46 + 692q−48−493q−50 + 241q−52−8q−54−149q−56 + 206q−58−199q−60 + 140q−62−72q−64 + 21q−66 + 16q−68−28q−70 + 27q−72−20q−74 + 10q−76−4q−78 + q−80
Further Quantum Invariants
K11a73 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["K11a73"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t4−7t3 + 21t2−37t + 45−37t−1 + 21t−2−7t−3 + t−4
In[5]:= Conway[K][z]
Out[5]= z8 + z6−z4 + 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]= { 177, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q5 + 5q4−11q3 + 18q2−25q + 29−28q−1 + 25q−2−18q−3 + 11q−4−5q−5 + q−6
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z8−2a2z6−z6a−2 + 4z6 + a4z4−5a2z4−2z4a−2 + 5z4 + a4z2−a2z2−a4 + 3a2 + a−2−2
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 3a2z10 + 3z10 + 9a3z9 + 19az9 + 10z9a−1 + 10a4z8 + 21a2z8 + 14z8a−2 + 25z8 + 5a5z7−8a3z7−23az7 + z7a−1 + 11z7a−3 + a6z6−21a4z6−60a2z6−18z6a−2 + 5z6a−4−61z6−9a5z5−14a3z5−16az5−26z5a−1−14z5a−3 + z5a−5−a6z4 + 13a4z4 + 42a2z4 + 6z4a−2−4z4a−4 + 38z4 + 4a5z3 + 14a3z3 + 23az3 + 18z3a−1 + 5z3a−3−2a4z2−5a2z2−3z2−2a3z−4az−2za−1−a4−3a2−a−2−2

[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["K11a73"];
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 + 21t2−37t + 45−37t−1 + 21t−2−7t−3 + t−4, −q5 + 5q4−11q3 + 18q2−25q + 29−28q−1 + 25q−2−18q−3 + 11q−4−5q−5 + q−6 }
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: (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 = 0 is the signature of K11a73. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-1012345χ
11           1-1
9          4 4
7         71 -6
5        114  7
3       147   -7
1      1511    4
-1     1415     1
-3    1114      -3
-5   714       7
-7  411        -7
-9 17         6
-11 4          -4
-131           1
Integral Khovanov Homology

(db, data source)

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

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K11a72

K11a74

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