K11n86

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K11n85.gif

K11n85

K11n87.gif

K11n87

Contents

K11n86.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n86 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X7,15,8,14 X9,16,10,17 X2,11,3,12 X13,19,14,18 X15,20,16,21 X17,1,18,22 X19,6,20,7 X21,9,22,8
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
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BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11n86 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:K11n86/ThurstonBennequinNumber
Hyperbolic Volume 11.7195
A-Polynomial See Data:K11n86/A-polynomial

[edit Notes for K11n86'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 K11n86's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+4 t^2-7 t+9-7 t^{-1} +4 t^{-2} - t^{-3}
Conway polynomial -z^6-2 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 33, 0 }
Jones polynomial q^6-3 q^5+4 q^4-5 q^3+6 q^2-5 q+5-3 q^{-1} + q^{-2}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -4 z^4 a^{-2} +z^4 a^{-4} +z^4-4 z^2 a^{-2} +2 z^2 a^{-4} +2 z^2+1
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +4 z^8 a^{-2} +3 z^8 a^{-4} +z^8-3 z^7 a^{-1} +3 z^7 a^{-5} -17 z^6 a^{-2} -11 z^6 a^{-4} +z^6 a^{-6} -5 z^6-11 z^5 a^{-3} -11 z^5 a^{-5} +22 z^4 a^{-2} +9 z^4 a^{-4} -3 z^4 a^{-6} +10 z^4+2 a z^3+7 z^3 a^{-1} +13 z^3 a^{-3} +8 z^3 a^{-5} -9 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} -5 z^2-a z-3 z a^{-1} -3 z a^{-3} -z a^{-5} +1
The A2 invariant q^6-q^4+q^2+2 q^{-4} +2 q^{-8} - q^{-10} - q^{-12} - q^{-16} + q^{-18}
The G2 invariant Data:K11n86/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: (0, 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
0 0 0 16 16 0 32 32 0 0 0 0 0 40 \frac{176}{3} -\frac{160}{3} \frac{104}{3} -8

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 K11n86. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456χ
13        11
11       2 -2
9      21 1
7     32  -1
5    32   1
3   23    1
1  33     0
-1 13      2
-3 2       -2
-51        1
Integral Khovanov Homology

(db, data source)

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

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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K11n85.gif

K11n85

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K11n87