K11n166

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K11n165.gif

K11n165

K11n167.gif

K11n167

Contents

K11n166.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n166 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X5,15,6,14 X18,7,19,8 X2,10,3,9 X20,11,21,12 X22,14,1,13 X15,5,16,4 X12,17,13,18 X8,19,9,20 X16,22,17,21
Gauss code 1, -5, 2, 8, -3, -1, 4, -10, 5, -2, 6, -9, 7, 3, -8, -11, 9, -4, 10, -6, 11, -7
Dowker-Thistlethwaite code 6 10 -14 18 2 20 22 -4 12 8 16
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation K11n166 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n166/ThurstonBennequinNumber
Hyperbolic Volume 14.1954
A-Polynomial See Data:K11n166/A-polynomial

[edit Notes for K11n166's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant -2

[edit Notes for K11n166's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+4 t^3-8 t^2+11 t-11+11 t^{-1} -8 t^{-2} +4 t^{-3} - t^{-4}
Conway polynomial -z^8-4 z^6-4 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 59, 2 }
Jones polynomial -q^6+4 q^5-7 q^4+9 q^3-10 q^2+10 q-8+6 q^{-1} -3 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-2} -6 z^6 a^{-2} +z^6 a^{-4} +z^6-13 z^4 a^{-2} +5 z^4 a^{-4} +4 z^4-13 z^2 a^{-2} +8 z^2 a^{-4} -z^2 a^{-6} +5 z^2-5 a^{-2} +4 a^{-4} - a^{-6} +3
Kauffman polynomial (db, data sources) 2 z^9 a^{-1} +2 z^9 a^{-3} +8 z^8 a^{-2} +4 z^8 a^{-4} +4 z^8+3 a z^7-z^7 a^{-1} -2 z^7 a^{-3} +2 z^7 a^{-5} +a^2 z^6-27 z^6 a^{-2} -14 z^6 a^{-4} -12 z^6-9 a z^5-9 z^5 a^{-1} -3 z^5 a^{-3} -3 z^5 a^{-5} -3 a^2 z^4+35 z^4 a^{-2} +26 z^4 a^{-4} +4 z^4 a^{-6} +10 z^4+6 a z^3+7 z^3 a^{-1} +7 z^3 a^{-3} +7 z^3 a^{-5} +z^3 a^{-7} +2 a^2 z^2-23 z^2 a^{-2} -17 z^2 a^{-4} -3 z^2 a^{-6} -7 z^2-a z-2 z a^{-1} -3 z a^{-3} -3 z a^{-5} -z a^{-7} +5 a^{-2} +4 a^{-4} + a^{-6} +3
The A2 invariant q^8-q^6+2 q^4+1+ q^{-2} -3 q^{-4} +2 q^{-6} -3 q^{-8} +2 q^{-10} +2 q^{-16} - q^{-18} + q^{-20} - q^{-22}
The G2 invariant Data:K11n166/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n124,}

Vassiliev invariants

V2 and V3: (-1, 0)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
-4 0 8 \frac{178}{3} \frac{110}{3} 0 96 32 32 -\frac{32}{3} 0 -\frac{712}{3} -\frac{440}{3} -\frac{5311}{30} \frac{5342}{15} -\frac{27422}{45} \frac{2047}{18} -\frac{5311}{30}

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (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=2 is the signature of K11n166. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-1012345χ
13         1-1
11        3 3
9       41 -3
7      53  2
5     54   -1
3    55    0
1   46     2
-1  24      -2
-3 14       3
-5 2        -2
-71         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

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K11n165.gif

K11n165

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K11n167