K11n98

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K11n97.gif

K11n97

K11n99.gif

K11n99

Contents

K11n98.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n98 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X5,14,6,15 X7,12,8,13 X18,9,19,10 X2,11,3,12 X13,6,14,7 X20,16,21,15 X22,18,1,17 X8,19,9,20 X16,22,17,21
Gauss code 1, -6, 2, -1, -3, 7, -4, -10, 5, -2, 6, 4, -7, 3, 8, -11, 9, -5, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 10 -14 -12 18 2 -6 20 22 8 16
A Braid Representative
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A Morse Link Presentation K11n98 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n98/ThurstonBennequinNumber
Hyperbolic Volume 13.759
A-Polynomial See Data:K11n98/A-polynomial

[edit Notes for K11n98'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 K11n98's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+7 t^2-16 t+21-16 t^{-1} +7 t^{-2} - t^{-3}
Conway polynomial -z^6+z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 69, 0 }
Jones polynomial q^4-4 q^3+7 q^2-10 q+12-11 q^{-1} +11 q^{-2} -7 q^{-3} +4 q^{-4} -2 q^{-5}
HOMFLY-PT polynomial (db, data sources) -z^6+3 a^2 z^4+z^4 a^{-2} -3 z^4-2 a^4 z^2+8 a^2 z^2+z^2 a^{-2} -4 z^2-3 a^4+6 a^2-2
Kauffman polynomial (db, data sources) a^3 z^9+a z^9+a^4 z^8+5 a^2 z^8+4 z^8+7 a z^7+7 z^7 a^{-1} -6 a^2 z^6+7 z^6 a^{-2} +z^6+3 a^5 z^5+3 a^3 z^5-11 a z^5-7 z^5 a^{-1} +4 z^5 a^{-3} -4 a^4 z^4-4 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} -9 z^4-8 a^5 z^3-13 a^3 z^3-3 a z^3-z^3 a^{-1} -3 z^3 a^{-3} +5 a^4 z^2+9 a^2 z^2+2 z^2 a^{-2} +6 z^2+5 a^5 z+9 a^3 z+6 a z+2 z a^{-1} -3 a^4-6 a^2-2
The A2 invariant Data:K11n98/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n98/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_78, K11n105,}

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

Vassiliev invariants

V2 and V3: (3, -3)
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 -24 72 126 26 -288 -496 -32 -152 288 288 1512 312 \frac{22991}{10} -\frac{4226}{15} \frac{6434}{5} \frac{155}{2} \frac{2191}{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=0 is the signature of K11n98. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-101234χ
9         11
7        3 -3
5       41 3
3      63  -3
1     64   2
-1    67    1
-3   55     0
-5  26      4
-7 25       -3
-9 2        2
-112         -2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-5 {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\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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K11n97.gif

K11n97

K11n99.gif

K11n99