K11n185

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K11n184.gif

K11n184

80px

K12a1

Contents

K11n185.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n185 at Knotilus!


Knot K11n185.
A graph, K11n185.
A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation X6271 X18,4,19,3 X16,5,17,6 X14,8,15,7 X9,21,10,20 X4,12,5,11 X2,13,3,14 X22,16,1,15 X12,18,13,17 X19,9,20,8 X21,11,22,10
Gauss code 1, -7, 2, -6, 3, -1, 4, 10, -5, 11, 6, -9, 7, -4, 8, -3, 9, -2, -10, 5, -11, -8
Dowker-Thistlethwaite code 6 18 16 14 -20 4 2 22 12 -8 -10
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11n185 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n185/ThurstonBennequinNumber
Hyperbolic Volume 17.1
A-Polynomial See Data:K11n185/A-polynomial

[edit Notes for K11n185's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n185's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+11 t^2-24 t+31-24 t^{-1} +11 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \left\{3,t^2+1\right\}
Determinant and Signature { 105, 4 }
Jones polynomial -q^{11}+5 q^{10}-10 q^9+14 q^8-18 q^7+18 q^6-16 q^5+13 q^4-7 q^3+3 q^2
HOMFLY-PT polynomial (db, data sources) -2 z^6 a^{-6} +3 z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} +7 z^2 a^{-4} -9 z^2 a^{-6} +5 z^2 a^{-8} -z^2 a^{-10} +4 a^{-4} -4 a^{-6} + a^{-8}
Kauffman polynomial (db, data sources) 3 z^9 a^{-7} +3 z^9 a^{-9} +6 z^8 a^{-6} +15 z^8 a^{-8} +9 z^8 a^{-10} +3 z^7 a^{-5} +7 z^7 a^{-7} +14 z^7 a^{-9} +10 z^7 a^{-11} -10 z^6 a^{-6} -24 z^6 a^{-8} -9 z^6 a^{-10} +5 z^6 a^{-12} -18 z^5 a^{-7} -35 z^5 a^{-9} -16 z^5 a^{-11} +z^5 a^{-13} +6 z^4 a^{-4} +16 z^4 a^{-6} +10 z^4 a^{-8} -5 z^4 a^{-10} -5 z^4 a^{-12} +z^3 a^{-5} +12 z^3 a^{-7} +16 z^3 a^{-9} +5 z^3 a^{-11} -9 z^2 a^{-4} -14 z^2 a^{-6} -3 z^2 a^{-8} +2 z^2 a^{-10} -3 z a^{-5} -3 z a^{-7} +z a^{-9} +z a^{-11} +4 a^{-4} +4 a^{-6} + a^{-8}
The A2 invariant 3 q^{-6} -2 q^{-8} +4 q^{-10} + q^{-12} -2 q^{-14} +4 q^{-16} -4 q^{-18} +2 q^{-20} -3 q^{-22} -2 q^{-24} +2 q^{-26} -3 q^{-28} +3 q^{-30} + q^{-32} - q^{-34}
The G2 invariant Data:K11n185/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_122,}

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{172}{3} \frac{68}{3} 128 \frac{736}{3} \frac{160}{3} 112 \frac{256}{3} 128 \frac{1376}{3} \frac{544}{3} \frac{14791}{15} -\frac{2148}{5} \frac{51244}{45} \frac{761}{9} \frac{2311}{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=4 is the signature of K11n185. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
0123456789χ
23         1-1
21        4 4
19       61 -5
17      84  4
15     106   -4
13    88    0
11   810     2
9  58      -3
7 28       6
515        -4
33         3
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=0 {\mathbb Z}^{3} {\mathbb Z}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=9 {\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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K11n184.gif

K11n184

80px

K12a1