K11n182

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K11n181

K11n183

Contents

Image:K11n182.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n182's page at Knotilus!

Visit K11n182's page at the original Knot Atlas!

[edit K11n182 Quick Notes]


[edit K11n182 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X6271 X3,12,4,13 X16,5,17,6 X18,8,19,7 X14,10,15,9 X11,4,12,5 X20,13,21,14 X22,16,1,15 X2,17,3,18 X8,20,9,19 X10,21,11,22
Gauss code 1, -9, -2, 6, 3, -1, 4, -10, 5, -11, -6, 2, 7, -5, 8, -3, 9, -4, 10, -7, 11, -8
Dowker-Thistlethwaite code 6 -12 16 18 14 -4 20 22 2 8 10
A Braid Representative
Image:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gif
A Morse Link Presentation Image:K11n182_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:K11n182/ThurstonBennequinNumber
Hyperbolic Volume 16.3897
A-Polynomial See Data:K11n182/A-polynomial

[edit Notes for K11n182'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 K11n182's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t4−5t3 + 11t2−18t + 23−18t−1 + 11t−2−5t−3 + t−4
Conway polynomial z8 + 3z6 + z4−3z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 93, 0 }
Jones polynomial −q5 + 4q4−8q3 + 13q2−15q + 16−15q−1 + 11q−2−7q−3 + 3q−4
HOMFLY-PT polynomial (db, data sources) z8−a2z6−z6a−2 + 5z6−4a2z4−3z4a−2 + 8z4 + a4z2−6a2z2−2z2a−2 + 4z2 + 2a4−3a2 + a−2 + 1
Kauffman polynomial (db, data sources) 3az9 + 3z9a−1 + 6a2z8 + 7z8a−2 + 13z8 + 3a3z7 + 2az7 + 6z7a−1 + 7z7a−3−14a2z6−9z6a−2 + 4z6a−4−27z6−7az5−19z5a−1−11z5a−3 + z5a−5 + 6a4z4 + 23a2z4−6z4a−4 + 23z4−5a3z3 + 2az3 + 12z3a−1 + 4z3a−3−z3a−5−9a4z2−18a2z2 + 2z2a−2 + 2z2a−4−9z2 + 3a3z + 2az−2za−1−za−3 + 2a4 + 3a2−a−2 + 1
The A2 invariant q16 + 3q12−2q10 + q8−q6−4q4 + 3q2−4 + 4q−2−q−4 + q−6 + 3q−8−2q−10 + 2q−12−q−14
The G2 invariant Data:K11n182/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n182 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["K11n182"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t4−5t3 + 11t2−18t + 23−18t−1 + 11t−2−5t−3 + t−4
In[5]:= Conway[K][z]
Out[5]= z8 + 3z6 + z4−3z2 + 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]= {1}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 93, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q5 + 4q4−8q3 + 13q2−15q + 16−15q−1 + 11q−2−7q−3 + 3q−4
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z8−a2z6−z6a−2 + 5z6−4a2z4−3z4a−2 + 8z4 + a4z2−6a2z2−2z2a−2 + 4z2 + 2a4−3a2 + a−2 + 1
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 3az9 + 3z9a−1 + 6a2z8 + 7z8a−2 + 13z8 + 3a3z7 + 2az7 + 6z7a−1 + 7z7a−3−14a2z6−9z6a−2 + 4z6a−4−27z6−7az5−19z5a−1−11z5a−3 + z5a−5 + 6a4z4 + 23a2z4−6z4a−4 + 23z4−5a3z3 + 2az3 + 12z3a−1 + 4z3a−3−z3a−5−9a4z2−18a2z2 + 2z2a−2 + 2z2a−4−9z2 + 3a3z + 2az−2za−1−za−3 + 2a4 + 3a2−a−2 + 1

[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["K11n182"];
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 + 11t2−18t + 23−18t−1 + 11t−2−5t−3 + t−4, −q5 + 4q4−8q3 + 13q2−15q + 16−15q−1 + 11q−2−7q−3 + 3q−4 }
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: (-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 K11n182. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-1012345χ
11         1-1
9        3 3
7       51 -4
5      83  5
3     75   -2
1    98    1
-1   78     1
-3  48      -4
-5 37       4
-7 4        -4
-93         3
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −1 i = 1
r = −4 {\mathbb Z}^{3}
r = −3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = −1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r = 1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r = 2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = 3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11n181

K11n183

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