K11a153

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K11a152.gif

K11a152

K11a154.gif

K11a154

Contents

K11a153.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a153 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X18,6,19,5 X12,8,13,7 X2,10,3,9 X8,12,9,11 X20,13,21,14 X22,15,1,16 X6,18,7,17 X16,19,17,20 X14,21,15,22
Gauss code 1, -5, 2, -1, 3, -9, 4, -6, 5, -2, 6, -4, 7, -11, 8, -10, 9, -3, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 10 18 12 2 8 20 22 6 16 14
A Braid Representative
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A Morse Link Presentation K11a153 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:K11a153/ThurstonBennequinNumber
Hyperbolic Volume 13.4113
A-Polynomial See Data:K11a153/A-polynomial

[edit Notes for K11a153's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11a153's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+10 t^2-20 t+25-20 t^{-1} +10 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-2 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 89, 0 }
Jones polynomial -q^7+2 q^6-5 q^5+9 q^4-11 q^3+14 q^2-14 q+13-10 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -z^6+a^2 z^4-2 z^4 a^{-2} +2 z^4 a^{-4} -3 z^4+2 a^2 z^2+5 z^2 a^{-4} -z^2 a^{-6} -4 z^2+a^2+ a^{-2} +3 a^{-4} -2 a^{-6} -2
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +5 z^9 a^{-3} +2 z^9 a^{-5} +5 z^8 a^{-2} +2 z^8 a^{-4} +2 z^8 a^{-6} +5 z^8+6 a z^7+2 z^7 a^{-1} -9 z^7 a^{-3} -4 z^7 a^{-5} +z^7 a^{-7} +5 a^2 z^6-11 z^6 a^{-2} -12 z^6 a^{-4} -8 z^6 a^{-6} -2 z^6+3 a^3 z^5-6 a z^5-12 z^5 a^{-1} -3 z^5 a^{-3} -5 z^5 a^{-5} -5 z^5 a^{-7} +a^4 z^4-5 a^2 z^4-2 z^4 a^{-2} +10 z^4 a^{-4} +10 z^4 a^{-6} -8 z^4-3 a^3 z^3+2 a z^3+7 z^3 a^{-1} +7 z^3 a^{-3} +13 z^3 a^{-5} +8 z^3 a^{-7} -a^4 z^2+3 a^2 z^2+5 z^2 a^{-2} -5 z^2 a^{-4} -5 z^2 a^{-6} +9 z^2+a^3 z+a z-z a^{-1} -3 z a^{-3} -6 z a^{-5} -4 z a^{-7} -a^2- a^{-2} +3 a^{-4} +2 a^{-6} -2
The A2 invariant Data:K11a153/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a153/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_92, K11a224, K11n35, K11n43,}

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

Vassiliev invariants

V2 and V3: (2, 5)
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 40 32 \frac{412}{3} \frac{92}{3} 320 \frac{1936}{3} \frac{160}{3} 168 \frac{256}{3} 800 \frac{3296}{3} \frac{736}{3} \frac{47311}{15} -\frac{5044}{15} \frac{76324}{45} \frac{1025}{9} \frac{3631}{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=0 is the signature of K11a153. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
15           1-1
13          1 1
11         41 -3
9        51  4
7       64   -2
5      85    3
3     66     0
1    78      -1
-1   47       3
-3  26        -4
-5 14         3
-7 2          -2
-91           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
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}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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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K11a152.gif

K11a152

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K11a154