K11a14

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K11a13

K11a15

Contents

Image:K11a14.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a14's page at Knotilus!

Visit K11a14's page at the original Knot Atlas!

[edit K11a14 Quick Notes]


[edit K11a14 Further Notes and Views]


[edit] Knot presentations

Planar diagram presentation X4251 X8493 X12,5,13,6 X2837 X14,9,15,10 X18,11,19,12 X6,13,7,14 X20,16,21,15 X10,17,11,18 X22,20,1,19 X16,22,17,21
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, 5, -9, 6, -3, 7, -5, 8, -11, 9, -6, 10, -8, 11, -10
Dowker-Thistlethwaite code 4 8 12 2 14 18 6 20 10 22 16
A Braid Representative
Image:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gif
A Morse Link Presentation Image:K11a14_ML.gif

[edit] Three dimensional invariants

Symmetry type Reversible
Unknotting number {1,2,3}
3-genus 4
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a14/ThurstonBennequinNumber
Hyperbolic Volume 16.4066
A-Polynomial See Data:K11a14/A-polynomial

[edit Notes for K11a14'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 K11a14's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial t4−5t3 + 15t2−28t + 35−28t−1 + 15t−2−5t−3 + t−4
Conway polynomial z8 + 3z6 + 5z4 + 3z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 133, 0 }
Jones polynomial q6−4q5 + 8q4−14q3 + 19q2−21q + 22−18q−1 + 14q−2−8q−3 + 3q−4−q−5
HOMFLY-PT polynomial (db, data sources) z8−a2z6−2z6a−2 + 6z6−4a2z4−8z4a−2 + z4a−4 + 16z4−7a2z2−12z2a−2 + 2z2a−4 + 20z2−4a2−6a−2 + a−4 + 10
Kauffman polynomial (db, data sources) z10a−2 + z10 + 5az9 + 9z9a−1 + 4z9a−3 + 8a2z8 + 17z8a−2 + 6z8a−4 + 19z8 + 6a3z7 + 4az7−z7a−1 + 5z7a−3 + 4z7a−5 + 3a4z6−14a2z6−46z6a−2−12z6a−4 + z6a−6−50z6 + a5z5−9a3z5−24az5−36z5a−1−32z5a−3−10z5a−5−4a4z4 + 17a2z4 + 41z4a−2 + 6z4a−4−2z4a−6 + 54z4−2a5z3 + 7a3z3 + 29az3 + 43z3a−1 + 31z3a−3 + 8z3a−5 + a4z2−13a2z2−21z2a−2−2z2a−4 + z2a−6−32z2 + a5z−3a3z−12az−16za−1−10za−3−2za−5 + 4a2 + 6a−2 + a−4 + 10
The A2 invariant −q14 + q12−4q10 + q8 + q6−2q4 + 7q2−1 + 5q−2−2q−6 + 2q−8−5q−10 + q−12−q−16 + q−18
The G2 invariant q80−2q78 + 5q76−8q74 + 10q72−10q70 + 4q68 + 10q66−29q64 + 52q62−74q60 + 78q58−61q56 + 7q54 + 89q52−198q50 + 290q48−316q46 + 225q44−31q42−245q40 + 508q38−649q36 + 586q34−312q32−108q30 + 511q28−748q26 + 712q24−413q22−30q20 + 427q18−607q16 + 494q14−126q12−308q10 + 628q8−659q6 + 373q4 + 140q2−660 + 1001q−2−986q−4 + 624q−6−15q−8−611q−10 + 1037q−12−1105q−14 + 803q−16−249q−18−340q−20 + 734q−22−798q−24 + 534q−26−76q−28−370q−30 + 589q−32−511q−34 + 163q−36 + 282q−38−631q−40 + 730q−42−540q−44 + 132q−46 + 313q−48−642q−50 + 740q−52−591q−54 + 284q−56 + 60q−58−328q−60 + 447q−62−411q−64 + 273q−66−95q−68−52q−70 + 134q−72−153q−74 + 123q−76−69q−78 + 24q−80 + 9q−82−23q−84 + 21q−86−16q−88 + 8q−90−3q−92 + q−94
Further Quantum Invariants
K11a14 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["K11a14"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t4−5t3 + 15t2−28t + 35−28t−1 + 15t−2−5t−3 + t−4
In[5]:= Conway[K][z]
Out[5]= z8 + 3z6 + 5z4 + 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]= { 133, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= q6−4q5 + 8q4−14q3 + 19q2−21q + 22−18q−1 + 14q−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 + 6z6−4a2z4−8z4a−2 + z4a−4 + 16z4−7a2z2−12z2a−2 + 2z2a−4 + 20z2−4a2−6a−2 + a−4 + 10
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z10a−2 + z10 + 5az9 + 9z9a−1 + 4z9a−3 + 8a2z8 + 17z8a−2 + 6z8a−4 + 19z8 + 6a3z7 + 4az7−z7a−1 + 5z7a−3 + 4z7a−5 + 3a4z6−14a2z6−46z6a−2−12z6a−4 + z6a−6−50z6 + a5z5−9a3z5−24az5−36z5a−1−32z5a−3−10z5a−5−4a4z4 + 17a2z4 + 41z4a−2 + 6z4a−4−2z4a−6 + 54z4−2a5z3 + 7a3z3 + 29az3 + 43z3a−1 + 31z3a−3 + 8z3a−5 + a4z2−13a2z2−21z2a−2−2z2a−4 + z2a−6−32z2 + a5z−3a3z−12az−16za−1−10za−3−2za−5 + 4a2 + 6a−2 + a−4 + 10

[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["K11a14"];
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 + 15t2−28t + 35−28t−1 + 15t−2−5t−3 + t−4, q6−4q5 + 8q4−14q3 + 19q2−21q + 22−18q−1 + 14q−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]= {}
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, -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 K11a14. 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         51 4
7        93  -6
5       105   5
3      119    -2
1     1110     1
-1    812      4
-3   610       -4
-5  28        6
-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}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = −1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r = 0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r = 1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r = 2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r = 3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r = 4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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K11a13

K11a15

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