K11n96

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K11n95.gif

K11n95

K11n97.gif

K11n97

Contents

K11n96.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n96 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X5,14,6,15 X7,13,8,12 X2,10,3,9 X11,21,12,20 X13,18,14,19 X15,22,16,1 X17,6,18,7 X19,9,20,8 X21,16,22,17
Gauss code 1, -5, 2, -1, -3, 9, -4, 10, 5, -2, -6, 4, -7, 3, -8, 11, -9, 7, -10, 6, -11, 8
Dowker-Thistlethwaite code 4 10 -14 -12 2 -20 -18 -22 -6 -8 -16
A Braid Representative
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A Morse Link Presentation K11n96 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n96/ThurstonBennequinNumber
Hyperbolic Volume 9.047
A-Polynomial See Data:K11n96/A-polynomial

[edit Notes for K11n96's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11n96's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-2 t^2+t+1+ t^{-1} -2 t^{-2} + t^{-3}
Conway polynomial z^6+4 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 7, 2 }
Jones polynomial -q^5+q^4+q-1+2 q^{-1} -2 q^{-2} +2 q^{-3} - q^{-4}
HOMFLY-PT polynomial (db, data sources) z^6-a^2 z^4+5 z^4-3 a^2 z^2-z^2 a^{-4} +6 z^2-a^2+ a^{-2} - a^{-4} +2
Kauffman polynomial (db, data sources) a z^9+z^9 a^{-1} +2 a^2 z^8+z^8 a^{-2} +3 z^8+a^3 z^7-4 a z^7-5 z^7 a^{-1} -11 a^2 z^6-6 z^6 a^{-2} +z^6 a^{-4} -18 z^6-5 a^3 z^5-a z^5+3 z^5 a^{-1} +z^5 a^{-5} +16 a^2 z^4+7 z^4 a^{-2} -5 z^4 a^{-4} +28 z^4+6 a^3 z^3+9 a z^3+4 z^3 a^{-1} -3 z^3 a^{-3} -4 z^3 a^{-5} -8 a^2 z^2-z^2 a^{-2} +4 z^2 a^{-4} -13 z^2-2 a^3 z-4 a z-2 z a^{-1} +2 z a^{-3} +2 z a^{-5} +a^2- a^{-2} - a^{-4} +2
The A2 invariant Data:K11n96/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n96/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: (2, 1)
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
8 8 32 \frac{76}{3} -\frac{52}{3} 64 \frac{272}{3} \frac{32}{3} 8 \frac{256}{3} 32 \frac{608}{3} -\frac{416}{3} \frac{4231}{15} \frac{932}{5} -\frac{10076}{45} \frac{233}{9} -\frac{1529}{15}

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 K11n96. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-1012345χ
11          1-1
9           0
7       111 1
5      11   0
3     111   1
1    231    0
-1   111     1
-3  121      0
-5 11        0
-7 1         1
-91          -1
Integral Khovanov Homology

(db, data source)

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

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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K11n95.gif

K11n95

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K11n97