K11n153

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K11n152.gif

K11n152

K11n154.gif

K11n154

Contents

K11n153.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n153 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X8394 X12,6,13,5 X20,8,21,7 X16,9,17,10 X18,11,19,12 X13,1,14,22 X4,16,5,15 X10,17,11,18 X2,19,3,20 X21,15,22,14
Gauss code 1, -10, 2, -8, 3, -1, 4, -2, 5, -9, 6, -3, -7, 11, 8, -5, 9, -6, 10, -4, -11, 7
Dowker-Thistlethwaite code 6 8 12 20 16 18 -22 4 10 2 -14
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11n153 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n153/ThurstonBennequinNumber
Hyperbolic Volume 13.0745
A-Polynomial See Data:K11n153/A-polynomial

[edit Notes for K11n153's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n153's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-4 t^3+7 t^2-10 t+13-10 t^{-1} +7 t^{-2} -4 t^{-3} + t^{-4}
Conway polynomial z^8+4 z^6+3 z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 57, 0 }
Jones polynomial 2 q^4-4 q^3+6 q^2-9 q+10-9 q^{-1} +8 q^{-2} -5 q^{-3} +3 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) z^8-a^2 z^6-z^6 a^{-2} +6 z^6-4 a^2 z^4-5 z^4 a^{-2} +12 z^4-4 a^2 z^2-8 z^2 a^{-2} +z^2 a^{-4} +9 z^2-4 a^{-2} +2 a^{-4} +3
Kauffman polynomial (db, data sources) 2 a z^9+2 z^9 a^{-1} +4 a^2 z^8+3 z^8 a^{-2} +7 z^8+4 a^3 z^7-3 a z^7-6 z^7 a^{-1} +z^7 a^{-3} +3 a^4 z^6-11 a^2 z^6-12 z^6 a^{-2} -26 z^6+a^5 z^5-9 a^3 z^5+11 z^5 a^{-1} +z^5 a^{-3} -7 a^4 z^4+12 a^2 z^4+25 z^4 a^{-2} +3 z^4 a^{-4} +41 z^4-2 a^5 z^3+4 a^3 z^3+5 a z^3-7 z^3 a^{-1} -6 z^3 a^{-3} +2 a^4 z^2-5 a^2 z^2-21 z^2 a^{-2} -7 z^2 a^{-4} -21 z^2-a^3 z-a z+3 z a^{-1} +3 z a^{-3} +4 a^{-2} +2 a^{-4} +3
The A2 invariant -q^{14}+q^{12}-q^{10}+2 q^8+q^6+3 q^2-2+2 q^{-2} -3 q^{-4} - q^{-6} - q^{-10} +2 q^{-12} + q^{-16}
The G2 invariant Data:K11n153/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (-2, -2)
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 -16 32 \frac{20}{3} -\frac{20}{3} 128 \frac{608}{3} \frac{224}{3} 16 -\frac{256}{3} 128 -\frac{160}{3} \frac{160}{3} \frac{4409}{15} -\frac{4196}{15} \frac{26276}{45} -\frac{953}{9} \frac{2009}{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 K11n153. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-101234χ
9         22
7        2 -2
5       42 2
3      52  -3
1     54   1
-1    56    1
-3   34     -1
-5  25      3
-7 13       -2
-9 2        2
-111         -1
Integral Khovanov Homology

(db, data source)

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

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K11n152.gif

K11n152

K11n154.gif

K11n154