K11n102

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K11n101.gif

K11n101

K11n103.gif

K11n103

Contents

K11n102.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

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Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X5,14,6,15 X7,12,8,13 X9,19,10,18 X2,11,3,12 X13,6,14,7 X15,22,16,1 X17,20,18,21 X19,9,20,8 X21,16,22,17
Gauss code 1, -6, 2, -1, -3, 7, -4, 10, -5, -2, 6, 4, -7, 3, -8, 11, -9, 5, -10, 9, -11, 8
Dowker-Thistlethwaite code 4 10 -14 -12 -18 2 -6 -22 -20 -8 -16
A Braid Representative
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A Morse Link Presentation K11n102 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n102/ThurstonBennequinNumber
Hyperbolic Volume 7.24432
A-Polynomial See Data:K11n102/A-polynomial

[edit Notes for K11n102's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n102's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^2+t+1+ t^{-1} - t^{-2}
Conway polynomial -z^4-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 3, -2 }
Jones polynomial q-1+2 q^{-1} - q^{-2} + q^{-3} - q^{-4} - q^{-5} + q^{-6} - q^{-7} + q^{-8}
HOMFLY-PT polynomial (db, data sources) a^8-z^2 a^6-a^6-z^4 a^2-3 z^2 a^2-a^2+z^2+2
Kauffman polynomial (db, data sources) a^7 z^9+a^5 z^9+a^8 z^8+2 a^6 z^8+a^4 z^8-7 a^7 z^7-7 a^5 z^7-7 a^8 z^6-14 a^6 z^6-7 a^4 z^6+15 a^7 z^5+14 a^5 z^5+a z^5+15 a^8 z^4+27 a^6 z^4+13 a^4 z^4+2 a^2 z^4+z^4-13 a^7 z^3-11 a^5 z^3-2 a z^3-11 a^8 z^2-15 a^6 z^2-6 a^4 z^2-5 a^2 z^2-3 z^2+5 a^7 z+6 a^5 z-a z+a^8+a^6+a^2+2
The A2 invariant Data:K11n102/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n102/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n38,}

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

Vassiliev invariants

V2 and V3: (-3, 6)
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 48 72 2 46 -576 -800 -192 -240 -288 1152 -24 -552 \frac{29649}{10} \frac{542}{5} \frac{4166}{5} \frac{709}{2} -\frac{591}{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=-2 is the signature of K11n102. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-8-7-6-5-4-3-2-1012χ
3          11
1           0
-1        21 1
-3      111  1
-5      11   0
-7    121    0
-9   1 1     -2
-11   11      0
-13 11        0
-15           0
-171          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1 i=1
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z} {\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}
r=2 {\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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K11n101.gif

K11n101

K11n103.gif

K11n103