K11a286

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K11a285.gif

K11a285

K11a287.gif

K11a287

Contents

K11a286.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a286 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X16,6,17,5 X18,7,19,8 X14,10,15,9 X2,11,3,12 X20,14,21,13 X4,16,5,15 X22,17,1,18 X12,20,13,19 X8,21,9,22
Gauss code 1, -6, 2, -8, 3, -1, 4, -11, 5, -2, 6, -10, 7, -5, 8, -3, 9, -4, 10, -7, 11, -9
Dowker-Thistlethwaite code 6 10 16 18 14 2 20 4 22 12 8
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation K11a286 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a286/ThurstonBennequinNumber
Hyperbolic Volume 17.8936
A-Polynomial See Data:K11a286/A-polynomial

[edit Notes for K11a286's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 1
Rasmussen s-Invariant -2

[edit Notes for K11a286's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+6 t^3-17 t^2+31 t-37+31 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4}
Conway polynomial -z^8-2 z^6-z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 147, 2 }
Jones polynomial q^7-4 q^6+9 q^5-16 q^4+21 q^3-23 q^2+24 q-20+15 q^{-1} -9 q^{-2} +4 q^{-3} - q^{-4}
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-10 z^4 a^{-2} +3 z^4 a^{-4} +7 z^4-2 a^2 z^2-8 z^2 a^{-2} +3 z^2 a^{-4} +8 z^2-a^2- a^{-2} +3
Kauffman polynomial (db, data sources) 3 z^{10} a^{-2} +3 z^{10}+6 a z^9+17 z^9 a^{-1} +11 z^9 a^{-3} +4 a^2 z^8+20 z^8 a^{-2} +17 z^8 a^{-4} +7 z^8+a^3 z^7-16 a z^7-41 z^7 a^{-1} -9 z^7 a^{-3} +15 z^7 a^{-5} -13 a^2 z^6-72 z^6 a^{-2} -31 z^6 a^{-4} +9 z^6 a^{-6} -45 z^6-3 a^3 z^5+8 a z^5+15 z^5 a^{-1} -21 z^5 a^{-3} -21 z^5 a^{-5} +4 z^5 a^{-7} +14 a^2 z^4+64 z^4 a^{-2} +20 z^4 a^{-4} -6 z^4 a^{-6} +z^4 a^{-8} +51 z^4+3 a^3 z^3+5 a z^3+11 z^3 a^{-1} +21 z^3 a^{-3} +11 z^3 a^{-5} -z^3 a^{-7} -6 a^2 z^2-21 z^2 a^{-2} -6 z^2 a^{-4} +z^2 a^{-6} -20 z^2-a^3 z-3 a z-5 z a^{-1} -4 z a^{-3} -z a^{-5} +a^2+ a^{-2} +3
The A2 invariant -q^{12}+q^{10}-2 q^6+4 q^4-3 q^2+2+2 q^{-2} -2 q^{-4} +6 q^{-6} -4 q^{-8} +3 q^{-10} -2 q^{-12} -3 q^{-14} +3 q^{-16} -2 q^{-18} + q^{-20}
The G2 invariant Data:K11a286/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a196, K11a216,}

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

Vassiliev invariants

V2 and V3: (1, 0)
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
4 0 8 \frac{62}{3} \frac{34}{3} 0 32 64 -32 \frac{32}{3} 0 \frac{248}{3} \frac{136}{3} \frac{3631}{30} \frac{1298}{15} -\frac{1378}{45} \frac{689}{18} \frac{271}{30}

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 K11a286. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123456χ
15           11
13          3 -3
11         61 5
9        103  -7
7       116   5
5      1210    -2
3     1211     1
1    913      4
-1   611       -5
-3  39        6
-5 16         -5
-7 3          3
-91           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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K11a285.gif

K11a285

K11a287.gif

K11a287