K11n147

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K11n146.gif

K11n146

K11n148.gif

K11n148

Contents

K11n147.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n147 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X14,4,15,3 X5,11,6,10 X7,20,8,21 X9,1,10,22 X11,19,12,18 X2,14,3,13 X15,9,16,8 X17,6,18,7 X19,13,20,12 X21,16,22,17
Gauss code 1, -7, 2, -1, -3, 9, -4, 8, -5, 3, -6, 10, 7, -2, -8, 11, -9, 6, -10, 4, -11, 5
Dowker-Thistlethwaite code 4 14 -10 -20 -22 -18 2 -8 -6 -12 -16
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation K11n147 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n147/ThurstonBennequinNumber
Hyperbolic Volume 12.6517
A-Polynomial See Data:K11n147/A-polynomial

[edit Notes for K11n147's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n147's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-4 t^3+7 t^2-5 t+3-5 t^{-1} +7 t^{-2} -4 t^{-3} + t^{-4}
Conway polynomial z^8+4 z^6+3 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 37, 4 }
Jones polynomial -2 q^7+4 q^6-5 q^5+6 q^4-6 q^3+6 q^2-4 q+3- q^{-1}
HOMFLY-PT polynomial (db, data sources) z^8 a^{-4} -z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -4 z^4 a^{-2} +12 z^4 a^{-4} -5 z^4 a^{-6} -3 z^2 a^{-2} +11 z^2 a^{-4} -6 z^2 a^{-6} +z^2 a^{-8} +3 a^{-4} -2 a^{-6}
Kauffman polynomial (db, data sources) 2 z^9 a^{-3} +2 z^9 a^{-5} +3 z^8 a^{-2} +7 z^8 a^{-4} +4 z^8 a^{-6} +z^7 a^{-1} -6 z^7 a^{-3} -5 z^7 a^{-5} +2 z^7 a^{-7} -14 z^6 a^{-2} -32 z^6 a^{-4} -18 z^6 a^{-6} -4 z^5 a^{-1} -2 z^5 a^{-3} -5 z^5 a^{-5} -7 z^5 a^{-7} +18 z^4 a^{-2} +41 z^4 a^{-4} +24 z^4 a^{-6} +z^4 a^{-8} +4 z^3 a^{-1} +9 z^3 a^{-3} +12 z^3 a^{-5} +7 z^3 a^{-7} -7 z^2 a^{-2} -19 z^2 a^{-4} -12 z^2 a^{-6} -z a^{-1} -3 z a^{-3} -5 z a^{-5} -2 z a^{-7} +z a^{-9} +3 a^{-4} +2 a^{-6}
The A2 invariant -q^2+1+ q^{-4} + q^{-6} + q^{-8} +2 q^{-10} -2 q^{-12} +2 q^{-14} - q^{-16} + q^{-18} - q^{-22} -2 q^{-26} + q^{-28}
The G2 invariant Data:K11n147/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (3, 5)
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
12 40 72 158 10 480 \frac{2320}{3} \frac{352}{3} 72 288 800 1896 120 \frac{37871}{10} \frac{1098}{5} \frac{17462}{15} \frac{17}{6} \frac{1231}{10}

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=4 is the signature of K11n147. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345χ
15        2-2
13       2 2
11      32 -1
9     32  1
7    33   0
5   33    0
3  24     2
1 12      -1
-1 2       2
-31        -1
Integral Khovanov Homology

(db, data source)

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

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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K11n146.gif

K11n146

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K11n148