K11n141

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K11n140.gif

K11n140

K11n142.gif

K11n142

Contents

K11n141.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n141 at Knotilus!

K11n141 is also known as the pretzel knot P(5,-3,-3).



Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X5,16,6,17 X7,14,8,15 X9,21,10,20 X11,19,12,18 X2,13,3,14 X15,6,16,7 X17,22,18,1 X19,11,20,10 X21,9,22,8
Gauss code 1, -7, 2, -1, -3, 8, -4, 11, -5, 10, -6, -2, 7, 4, -8, 3, -9, 6, -10, 5, -11, 9
Dowker-Thistlethwaite code 4 12 -16 -14 -20 -18 2 -6 -22 -10 -8
A Braid Representative
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A Morse Link Presentation K11n141 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2,3\}
3-genus 1
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n141/ThurstonBennequinNumber
Hyperbolic Volume 7.80349
A-Polynomial See Data:K11n141/A-polynomial

[edit Notes for K11n141's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 1
Rasmussen s-Invariant 0

[edit Notes for K11n141's four dimensional invariants]

Polynomial invariants

Alexander polynomial -5 t+11-5 t^{-1}
Conway polynomial 1-5 z^2
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 21, 0 }
Jones polynomial 2 q^2-2 q+3-4 q^{-1} +3 q^{-2} -3 q^{-3} +2 q^{-4} - q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) a^6-z^2 a^4-2 z^2 a^2-a^2-2 z^2-1+2 a^{-2}
Kauffman polynomial (db, data sources) a^3 z^9+a z^9+a^4 z^8+2 a^2 z^8+z^8+a^5 z^7-6 a^3 z^7-7 a z^7+a^6 z^6-4 a^4 z^6-11 a^2 z^6-6 z^6-4 a^5 z^5+15 a^3 z^5+20 a z^5+z^5 a^{-1} -5 a^6 z^4+4 a^4 z^4+22 a^2 z^4+13 z^4+3 a^5 z^3-19 a^3 z^3-24 a z^3-2 z^3 a^{-1} +6 a^6 z^2-2 a^4 z^2-16 a^2 z^2+2 z^2 a^{-2} -6 z^2+8 a^3 z+11 a z+3 z a^{-1} -a^6+a^2-2 a^{-2} -1
The A2 invariant Data:K11n141/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n141/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: (-5, 4)
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
-20 32 200 \frac{746}{3} \frac{310}{3} -640 -\frac{3040}{3} -\frac{640}{3} -224 -\frac{4000}{3} 512 -\frac{14920}{3} -\frac{6200}{3} -\frac{19279}{6} 778 -\frac{30782}{9} \frac{9995}{18} -\frac{5071}{6}

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=0 is the signature of K11n141. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012χ
5        22
3         0
1      32 1
-1     21  -1
-3    12   -1
-5   22    0
-7   1     -1
-9 12      1
-11         0
-131        1
Integral Khovanov Homology

(db, data source)

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

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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K11n140.gif

K11n140

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K11n142