K11n85

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K11n84.gif

K11n84

K11n86.gif

K11n86

Contents

K11n85.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n85 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X7,15,8,14 X16,10,17,9 X2,11,3,12 X13,19,14,18 X20,16,21,15 X22,17,1,18 X19,6,20,7 X8,21,9,22
Gauss code 1, -6, 2, -1, 3, 10, -4, -11, 5, -2, 6, -3, -7, 4, 8, -5, 9, 7, -10, -8, 11, -9
Dowker-Thistlethwaite code 4 10 12 -14 16 2 -18 20 22 -6 8
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation K11n85 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n85/ThurstonBennequinNumber
Hyperbolic Volume 11.7267
A-Polynomial See Data:K11n85/A-polynomial

[edit Notes for K11n85's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 1
Rasmussen s-Invariant 0

[edit Notes for K11n85's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+5 t^2-10 t+13-10 t^{-1} +5 t^{-2} - t^{-3}
Conway polynomial -z^6-z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 45, 0 }
Jones polynomial q^6-3 q^5+4 q^4-6 q^3+8 q^2-7 q+7-5 q^{-1} +3 q^{-2} - q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -4 z^4 a^{-2} +z^4 a^{-4} +2 z^4-a^2 z^2-5 z^2 a^{-2} +2 z^2 a^{-4} +5 z^2-a^2- a^{-2} +3
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +5 z^8 a^{-2} +3 z^8 a^{-4} +2 z^8+a z^7+2 z^7 a^{-3} +3 z^7 a^{-5} -18 z^6 a^{-2} -10 z^6 a^{-4} +z^6 a^{-6} -7 z^6-a z^5-6 z^5 a^{-1} -16 z^5 a^{-3} -11 z^5 a^{-5} +3 a^2 z^4+23 z^4 a^{-2} +8 z^4 a^{-4} -3 z^4 a^{-6} +15 z^4+a^3 z^3+4 a z^3+11 z^3 a^{-1} +17 z^3 a^{-3} +9 z^3 a^{-5} -3 a^2 z^2-12 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} -11 z^2-a^3 z-3 a z-5 z a^{-1} -4 z a^{-3} -z a^{-5} +a^2+ a^{-2} +3
The A2 invariant -q^{10}+q^6-q^4+2 q^2+ q^{-2} +2 q^{-4} +2 q^{-8} -2 q^{-10} - q^{-12} - q^{-16} + q^{-18}
The G2 invariant Data:K11n85/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_18, 9_24, K11n164,}

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

Vassiliev invariants

V2 and V3: (1, 0)
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 0 8 \frac{62}{3} \frac{34}{3} 0 32 32 0 \frac{32}{3} 0 \frac{248}{3} \frac{136}{3} \frac{3631}{30} \frac{818}{15} \frac{1502}{45} \frac{113}{18} \frac{271}{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=0 is the signature of K11n85. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-10123456χ
13         11
11        2 -2
9       21 1
7      42  -2
5     42   2
3    34    1
1   44     0
-1  24      2
-3 13       -2
-5 2        2
-71         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\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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K11n84.gif

K11n84

K11n86.gif

K11n86