K11n151

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K11n150.gif

K11n150

K11n152.gif

K11n152

Contents

K11n151.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n151 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X8493 X12,5,13,6 X2837 X9,16,10,17 X11,19,12,18 X4,13,5,14 X15,21,16,20 X17,22,18,1 X19,15,20,14 X21,11,22,10
Gauss code 1, -4, 2, -7, 3, -1, 4, -2, -5, 11, -6, -3, 7, 10, -8, 5, -9, 6, -10, 8, -11, 9
Dowker-Thistlethwaite code 6 8 12 2 -16 -18 4 -20 -22 -14 -10
A Braid Representative
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A Morse Link Presentation K11n151 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n151/ThurstonBennequinNumber
Hyperbolic Volume 12.4339
A-Polynomial See Data:K11n151/A-polynomial

[edit Notes for K11n151'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 K11n151's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^2+6 t-7+6 t^{-1} -2 t^{-2}
Conway polynomial -2 z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 23, 2 }
Jones polynomial -q^8+3 q^7-4 q^6+5 q^5-5 q^4+4 q^3-3 q^2+q+1- q^{-1} + q^{-2}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} +z^6 a^{-4} -7 z^4 a^{-2} +5 z^4 a^{-4} -z^4 a^{-6} +z^4-13 z^2 a^{-2} +9 z^2 a^{-4} -2 z^2 a^{-6} +4 z^2-7 a^{-2} +5 a^{-4} - a^{-6} +4
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +2 z^8 a^{-2} +2 z^8 a^{-4} +z^8 a^{-6} +z^8-7 z^7 a^{-1} -7 z^7 a^{-3} +3 z^7 a^{-5} +3 z^7 a^{-7} -17 z^6 a^{-2} -13 z^6 a^{-4} +3 z^6 a^{-8} -7 z^6+13 z^5 a^{-1} +11 z^5 a^{-3} -11 z^5 a^{-5} -8 z^5 a^{-7} +z^5 a^{-9} +40 z^4 a^{-2} +28 z^4 a^{-4} -5 z^4 a^{-6} -8 z^4 a^{-8} +15 z^4-7 z^3 a^{-1} +2 z^3 a^{-3} +15 z^3 a^{-5} +4 z^3 a^{-7} -2 z^3 a^{-9} -31 z^2 a^{-2} -19 z^2 a^{-4} +2 z^2 a^{-6} +3 z^2 a^{-8} -13 z^2-4 z a^{-3} -6 z a^{-5} -2 z a^{-7} +7 a^{-2} +5 a^{-4} + a^{-6} +4
The A2 invariant q^6+q^4+2 q^2+ q^{-2} -2 q^{-4} -2 q^{-6} -2 q^{-10} +2 q^{-12} +2 q^{-16} + q^{-18} - q^{-20} + q^{-22} - q^{-24}
The G2 invariant Data:K11n151/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_6, K11n20, K11n152,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n152,}

Vassiliev invariants

V2 and V3: (-2, -1)
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
-8 -8 32 \frac{212}{3} \frac{100}{3} 64 \frac{400}{3} \frac{64}{3} 24 -\frac{256}{3} 32 -\frac{1696}{3} -\frac{800}{3} -\frac{8791}{15} \frac{1604}{15} -\frac{29044}{45} \frac{871}{9} -\frac{2071}{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 K11n151. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          2 2
13         21 -1
11        32  1
9      132   0
7      23    -1
5    133     1
3   112      -2
1   13       2
-1 11         0
-3            0
-51           1
Integral Khovanov Homology

(db, data source)

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

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K11n150.gif

K11n150

K11n152.gif

K11n152