K11n150

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K11n149.gif

K11n149

K11n151.gif

K11n151

Contents

K11n150.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n150 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X8394 X10,6,11,5 X18,8,19,7 X16,9,17,10 X11,1,12,22 X13,21,14,20 X4,16,5,15 X2,17,3,18 X19,15,20,14 X21,13,22,12
Gauss code 1, -9, 2, -8, 3, -1, 4, -2, 5, -3, -6, 11, -7, 10, 8, -5, 9, -4, -10, 7, -11, 6
Dowker-Thistlethwaite code 6 8 10 18 16 -22 -20 4 2 -14 -12
A Braid Representative
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A Morse Link Presentation K11n150 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:K11n150/ThurstonBennequinNumber
Hyperbolic Volume 14.7873
A-Polynomial See Data:K11n150/A-polynomial

[edit Notes for K11n150's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for K11n150's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-9 t^2+17 t-19+17 t^{-1} -9 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+3 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 75, 2 }
Jones polynomial 2 q^7-5 q^6+8 q^5-12 q^4+13 q^3-12 q^2+11 q-7+4 q^{-1} - q^{-2}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -z^4+3 z^2 a^{-4} -3 z^2 a^{-6} -z^2+ a^{-4} -2 a^{-6} + a^{-8} +1
Kauffman polynomial (db, data sources) 2 z^9 a^{-3} +2 z^9 a^{-5} +5 z^8 a^{-2} +8 z^8 a^{-4} +3 z^8 a^{-6} +6 z^7 a^{-1} +3 z^7 a^{-3} -2 z^7 a^{-5} +z^7 a^{-7} -6 z^6 a^{-2} -17 z^6 a^{-4} -7 z^6 a^{-6} +4 z^6+a z^5-10 z^5 a^{-1} -6 z^5 a^{-3} +8 z^5 a^{-5} +3 z^5 a^{-7} +19 z^4 a^{-4} +15 z^4 a^{-6} +3 z^4 a^{-8} -7 z^4-a z^3+2 z^3 a^{-1} -2 z^3 a^{-3} -12 z^3 a^{-5} -7 z^3 a^{-7} -z^2 a^{-2} -10 z^2 a^{-4} -12 z^2 a^{-6} -5 z^2 a^{-8} +2 z^2+3 z a^{-3} +7 z a^{-5} +4 z a^{-7} + a^{-4} +2 a^{-6} + a^{-8} +1
The A2 invariant Data:K11n150/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n150/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a258,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n66,}

Vassiliev invariants

V2 and V3: (-1, -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
-4 -24 8 -\frac{254}{3} -\frac{58}{3} 96 -112 0 -56 -\frac{32}{3} 288 \frac{1016}{3} \frac{232}{3} \frac{16049}{30} \frac{982}{15} \frac{11578}{45} -\frac{1937}{18} \frac{1649}{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 K11n150. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-10123456χ
15         22
13        3 -3
11       52 3
9      73  -4
7     65   1
5    67    1
3   56     -1
1  37      4
-1 14       -3
-3 3        3
-51         -1
Integral Khovanov Homology

(db, data source)

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

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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K11n149.gif

K11n149

K11n151.gif

K11n151