K11n91

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K11n90.gif

K11n90

K11n92.gif

K11n92

Contents

K11n91.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 X12,6,13,5 X7,16,8,17 X9,18,10,19 X2,11,3,12 X20,13,21,14 X15,8,16,9 X17,6,18,7 X22,19,1,20 X14,21,15,22
Gauss code 1, -6, 2, -1, 3, 9, -4, 8, -5, -2, 6, -3, 7, -11, -8, 4, -9, 5, 10, -7, 11, -10
Dowker-Thistlethwaite code 4 10 12 -16 -18 2 20 -8 -6 22 14
A Braid Representative
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A Morse Link Presentation K11n91 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:K11n91/ThurstonBennequinNumber
Hyperbolic Volume 9.90352
A-Polynomial See Data:K11n91/A-polynomial

[edit Notes for K11n91'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 K11n91's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^2+8 t-13+8 t^{-1} - t^{-2}
Conway polynomial -z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \left\{2,t^2+t+1\right\}
Determinant and Signature { 31, -2 }
Jones polynomial  q^{-1} -2 q^{-2} +4 q^{-3} -4 q^{-4} +5 q^{-5} -5 q^{-6} +4 q^{-7} -3 q^{-8} +2 q^{-9} - q^{-10}
HOMFLY-PT polynomial (db, data sources) -a^{10}+2 z^2 a^8+2 a^8-z^4 a^6-2 z^2 a^6-3 a^6+3 z^2 a^4+3 a^4+z^2 a^2
Kauffman polynomial (db, data sources) z^7 a^{11}-5 z^5 a^{11}+7 z^3 a^{11}-3 z a^{11}+2 z^8 a^{10}-10 z^6 a^{10}+14 z^4 a^{10}-6 z^2 a^{10}+a^{10}+z^9 a^9-2 z^7 a^9-6 z^5 a^9+11 z^3 a^9-3 z a^9+4 z^8 a^8-18 z^6 a^8+23 z^4 a^8-11 z^2 a^8+2 a^8+z^9 a^7-2 z^7 a^7-2 z^5 a^7+2 z^3 a^7-z a^7+2 z^8 a^6-8 z^6 a^6+13 z^4 a^6-12 z^2 a^6+3 a^6+z^7 a^5-z^5 a^5-z a^5+4 z^4 a^4-6 z^2 a^4+3 a^4+2 z^3 a^3+z^2 a^2
The A2 invariant Data:K11n91/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n91/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: (4, -9)
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
16 -72 128 \frac{1208}{3} \frac{208}{3} -1152 -2416 -416 -392 \frac{2048}{3} 2592 \frac{19328}{3} \frac{3328}{3} \frac{219182}{15} -\frac{1376}{5} \frac{299408}{45} \frac{1378}{9} \frac{13742}{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 K11n91. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-9-8-7-6-5-4-3-2-10χ
-1         11
-3        21-1
-5       2  2
-7      22  0
-9     32   1
-11    22    0
-13   23     -1
-15  12      1
-17 12       -1
-19 1        1
-211         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z} {\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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K11n90.gif

K11n90

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K11n92