K11a175

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K11a174

K11a176

Contents

Image:K11a175.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a175's page at Knotilus!

Visit K11a175's page at the original Knot Atlas!

[edit K11a175 Quick Notes]


[edit K11a175 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X14,5,15,6 X16,7,17,8 X18,10,19,9 X2,12,3,11 X22,13,1,14 X6,15,7,16 X20,18,21,17 X10,20,11,19 X8,21,9,22
Gauss code 1, -6, 2, -1, 3, -8, 4, -11, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, -9, 11, -7
Dowker-Thistlethwaite code 4 12 14 16 18 2 22 6 20 10 8
A Braid Representative
Image:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart4.gif
A Morse Link Presentation Image:K11a175_ML.gif

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a175/ThurstonBennequinNumber
Hyperbolic Volume 13.9605
A-Polynomial See Data:K11a175/A-polynomial

[edit Notes for K11a175'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 K11a175's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t4−5t3 + 13t2−21t + 25−21t−1 + 13t−2−5t−3 + t−4
Conway polynomial z8 + 3z6 + 3z4 + 2z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 105, 0 }
Jones polynomial q6−4q5 + 7q4−11q3 + 15q2−16q + 17−14q−1 + 10q−2−6q−3 + 3q−4−q−5
HOMFLY-PT polynomial (db, data sources) z8−a2z6−2z6a−2 + 6z6−4a2z4−8z4a−2 + z4a−4 + 14z4−5a2z2−9z2a−2 + 2z2a−4 + 14z2−2a2−2a−2 + 5
Kauffman polynomial (db, data sources) z10a−2 + z10 + 3az9 + 7z9a−1 + 4z9a−3 + 4a2z8 + 10z8a−2 + 6z8a−4 + 8z8 + 4a3z7−12z7a−1−4z7a−3 + 4z7a−5 + 3a4z6−3a2z6−35z6a−2−17z6a−4 + z6a−6−23z6 + a5z5−5a3z5−3az5 + 3z5a−1−11z5a−3−11z5a−5−6a4z4 + 39z4a−2 + 13z4a−4−2z4a−6 + 30z4−2a5z3 + 2az3 + 7z3a−1 + 13z3a−3 + 6z3a−5 + 3a4z2−3a2z2−17z2a−2−4z2a−4−19z2 + a5z + a3z−az−3za−1−2za−3 + 2a2 + 2a−2 + 5
The A2 invariant −q14 + q12−2q10 + q8 + q6−2q4 + 4q2−2 + 3q−2 + q−4 + 3q−8−3q−10−q−14−q−16 + q−18
The G2 invariant q80−2q78 + 5q76−8q74 + 8q72−6q70−2q68 + 16q66−28q64 + 40q62−44q60 + 31q58−7q56−33q54 + 73q52−105q50 + 116q48−99q46 + 48q44 + 28q42−114q40 + 187q38−217q36 + 188q34−106q32−21q30 + 149q28−237q26 + 260q24−186q22 + 57q20 + 82q18−179q16 + 186q14−106q12−28q10 + 153q8−206q6 + 154q4−3q2−182 + 332q−2−366q−4 + 266q−6−62q−8−177q−10 + 371q−12−440q−14 + 374q−16−184q−18−46q−20 + 244q−22−332q−24 + 289q−26−145q−28−33q−30 + 168q−32−213q−34 + 144q−36 + 5q−38−159q−40 + 256q−42−247q−44 + 122q−46 + 54q−48−226q−50 + 318q−52−304q−54 + 192q−56−27q−58−131q−60 + 228q−62−241q−64 + 183q−66−85q−68−13q−70 + 77q−72−103q−74 + 90q−76−55q−78 + 24q−80 + 5q−82−17q−84 + 17q−86−14q−88 + 7q−90−3q−92 + q−94
Further Quantum Invariants
K11a175 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["K11a175"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t4−5t3 + 13t2−21t + 25−21t−1 + 13t−2−5t−3 + t−4
In[5]:= Conway[K][z]
Out[5]= z8 + 3z6 + 3z4 + 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]= { 105, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q6−4q5 + 7q4−11q3 + 15q2−16q + 17−14q−1 + 10q−2−6q−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 + 6z6−4a2z4−8z4a−2 + z4a−4 + 14z4−5a2z2−9z2a−2 + 2z2a−4 + 14z2−2a2−2a−2 + 5
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z10a−2 + z10 + 3az9 + 7z9a−1 + 4z9a−3 + 4a2z8 + 10z8a−2 + 6z8a−4 + 8z8 + 4a3z7−12z7a−1−4z7a−3 + 4z7a−5 + 3a4z6−3a2z6−35z6a−2−17z6a−4 + z6a−6−23z6 + a5z5−5a3z5−3az5 + 3z5a−1−11z5a−3−11z5a−5−6a4z4 + 39z4a−2 + 13z4a−4−2z4a−6 + 30z4−2a5z3 + 2az3 + 7z3a−1 + 13z3a−3 + 6z3a−5 + 3a4z2−3a2z2−17z2a−2−4z2a−4−19z2 + a5z + a3z−az−3za−1−2za−3 + 2a2 + 2a−2 + 5

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

Same Alexander/Conway Polynomial: {K11a306,}

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["K11a175"];
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 + 13t2−21t + 25−21t−1 + 13t−2−5t−3 + t−4, q6−4q5 + 7q4−11q3 + 15q2−16q + 17−14q−1 + 10q−2−6q−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]= {K11a306,}
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, 0)

[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 K11a175. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-10123456χ
13           11
11          3 -3
9         41 3
7        73  -4
5       84   4
3      87    -1
1     98     1
-1    69      3
-3   48       -4
-5  26        4
-7 14         -3
-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}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = −1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r = 1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = 2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = 3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11a174

K11a176

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