K11n160

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K11n159.gif

K11n159

K11n161.gif

K11n161

Contents

K11n160.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n160 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X3,11,4,10 X12,6,13,5 X20,8,21,7 X16,10,17,9 X11,19,12,18 X22,13,1,14 X8,16,9,15 X17,4,18,5 X2,19,3,20 X14,21,15,22
Gauss code 1, -10, -2, 9, 3, -1, 4, -8, 5, 2, -6, -3, 7, -11, 8, -5, -9, 6, 10, -4, 11, -7
Dowker-Thistlethwaite code 6 -10 12 20 16 -18 22 8 -4 2 14
A Braid Representative
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BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gif
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A Morse Link Presentation K11n160 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n160/ThurstonBennequinNumber
Hyperbolic Volume 14.3018
A-Polynomial See Data:K11n160/A-polynomial

[edit Notes for K11n160'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 K11n160's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-6 t^2+16 t-21+16 t^{-1} -6 t^{-2} + t^{-3}
Conway polynomial z^6+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 67, 2 }
Jones polynomial -q^8+3 q^7-6 q^6+9 q^5-11 q^4+12 q^3-10 q^2+8 q-5+2 q^{-1}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-4} -3 z^4 a^{-2} +4 z^4 a^{-4} -z^4 a^{-6} -7 z^2 a^{-2} +8 z^2 a^{-4} -2 z^2 a^{-6} +2 z^2-4 a^{-2} +5 a^{-4} -2 a^{-6} +2
Kauffman polynomial (db, data sources) 2 z^9 a^{-3} +2 z^9 a^{-5} +3 z^8 a^{-2} +8 z^8 a^{-4} +5 z^8 a^{-6} +z^7 a^{-1} -3 z^7 a^{-3} +z^7 a^{-5} +5 z^7 a^{-7} -8 z^6 a^{-2} -24 z^6 a^{-4} -13 z^6 a^{-6} +3 z^6 a^{-8} +3 z^5 a^{-1} +6 z^5 a^{-3} -10 z^5 a^{-5} -12 z^5 a^{-7} +z^5 a^{-9} +17 z^4 a^{-2} +35 z^4 a^{-4} +15 z^4 a^{-6} -6 z^4 a^{-8} +3 z^4-7 z^3 a^{-1} -4 z^3 a^{-3} +14 z^3 a^{-5} +9 z^3 a^{-7} -2 z^3 a^{-9} -17 z^2 a^{-2} -21 z^2 a^{-4} -8 z^2 a^{-6} +z^2 a^{-8} -5 z^2+z a^{-1} -4 z a^{-5} -3 z a^{-7} +4 a^{-2} +5 a^{-4} +2 a^{-6} +2
The A2 invariant 2 q^4-1+2 q^{-2} -3 q^{-4} + q^{-6} +3 q^{-12} - q^{-14} +3 q^{-16} - q^{-18} -2 q^{-20} + q^{-22} - q^{-24}
The G2 invariant Data:K11n160/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n7, K11n131,}

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

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{206}{3} \frac{10}{3} 96 240 -32 88 \frac{32}{3} 288 \frac{824}{3} \frac{40}{3} \frac{31711}{30} -\frac{5462}{15} \frac{32582}{45} \frac{1025}{18} \frac{2911}{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 K11n160. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-101234567χ
17         1-1
15        2 2
13       41 -3
11      52  3
9     64   -2
7    65    1
5   46     2
3  46      -2
1 25       3
-1 3        -3
-32         2
Integral Khovanov Homology

(db, data source)

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

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K11n159.gif

K11n159

K11n161.gif

K11n161