K11a227

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K11a226.gif

K11a226

K11a228.gif

K11a228

Contents

K11a227.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a227 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X18,6,19,5 X22,8,1,7 X16,10,17,9 X2,12,3,11 X8,14,9,13 X20,16,21,15 X10,18,11,17 X6,20,7,19 X14,22,15,21
Gauss code 1, -6, 2, -1, 3, -10, 4, -7, 5, -9, 6, -2, 7, -11, 8, -5, 9, -3, 10, -8, 11, -4
Dowker-Thistlethwaite code 4 12 18 22 16 2 8 20 10 6 14
A Braid Representative
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A Morse Link Presentation K11a227 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a227/ThurstonBennequinNumber
Hyperbolic Volume 17.3495
A-Polynomial See Data:K11a227/A-polynomial

[edit Notes for K11a227's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -6

[edit Notes for K11a227's four dimensional invariants]

Polynomial invariants

Alexander polynomial 5 t^3-17 t^2+31 t-37+31 t^{-1} -17 t^{-2} +5 t^{-3}
Conway polynomial 5 z^6+13 z^4+8 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 143, 6 }
Jones polynomial -q^{14}+4 q^{13}-9 q^{12}+15 q^{11}-21 q^{10}+23 q^9-23 q^8+20 q^7-14 q^6+9 q^5-3 q^4+q^3
HOMFLY-PT polynomial (db, data sources) z^6 a^{-6} +3 z^6 a^{-8} +z^6 a^{-10} +3 z^4 a^{-6} +11 z^4 a^{-8} -z^4 a^{-12} +3 z^2 a^{-6} +12 z^2 a^{-8} -6 z^2 a^{-10} -z^2 a^{-12} + a^{-6} +4 a^{-8} -5 a^{-10} + a^{-12}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-10} +2 z^{10} a^{-12} +5 z^9 a^{-9} +12 z^9 a^{-11} +7 z^9 a^{-13} +6 z^8 a^{-8} +12 z^8 a^{-10} +16 z^8 a^{-12} +10 z^8 a^{-14} +3 z^7 a^{-7} -z^7 a^{-9} -13 z^7 a^{-11} -z^7 a^{-13} +8 z^7 a^{-15} +z^6 a^{-6} -14 z^6 a^{-8} -34 z^6 a^{-10} -38 z^6 a^{-12} -15 z^6 a^{-14} +4 z^6 a^{-16} -6 z^5 a^{-7} -18 z^5 a^{-9} -14 z^5 a^{-11} -15 z^5 a^{-13} -12 z^5 a^{-15} +z^5 a^{-17} -3 z^4 a^{-6} +15 z^4 a^{-8} +29 z^4 a^{-10} +25 z^4 a^{-12} +9 z^4 a^{-14} -5 z^4 a^{-16} +3 z^3 a^{-7} +20 z^3 a^{-9} +24 z^3 a^{-11} +15 z^3 a^{-13} +7 z^3 a^{-15} -z^3 a^{-17} +3 z^2 a^{-6} -12 z^2 a^{-8} -16 z^2 a^{-10} -4 z^2 a^{-12} -2 z^2 a^{-14} +z^2 a^{-16} -8 z a^{-9} -10 z a^{-11} -4 z a^{-13} -2 z a^{-15} - a^{-6} +4 a^{-8} +5 a^{-10} + a^{-12}
The A2 invariant Data:K11a227/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a227/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: (8, 21)
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
32 168 512 \frac{3424}{3} \frac{440}{3} 5376 8688 1440 936 \frac{16384}{3} 14112 \frac{109568}{3} \frac{14080}{3} \frac{1024804}{15} \frac{61144}{15} \frac{992536}{45} \frac{3884}{9} \frac{39364}{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=6 is the signature of K11a227. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
29           1-1
27          3 3
25         61 -5
23        93  6
21       126   -6
19      119    2
17     1212     0
15    811      -3
13   612       6
11  38        -5
9  6         6
713          -2
51           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=5 i=7
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}^{3}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=6 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=7 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=8 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=9 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=11 {\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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K11a226.gif

K11a226

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K11a228