K11n184

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K11n183.gif

K11n183

K11n185.gif

K11n185

Contents

K11n184.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n184 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X3,15,4,14 X10,6,11,5 X18,7,19,8 X2,10,3,9 X22,11,1,12 X20,14,21,13 X15,5,16,4 X12,18,13,17 X8,19,9,20 X16,22,17,21
Gauss code 1, -5, -2, 8, 3, -1, 4, -10, 5, -3, 6, -9, 7, 2, -8, -11, 9, -4, 10, -7, 11, -6
Dowker-Thistlethwaite code 6 -14 10 18 2 22 20 -4 12 8 16
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11n184 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n184/ThurstonBennequinNumber
Hyperbolic Volume 16.384
A-Polynomial See Data:K11n184/A-polynomial

[edit Notes for K11n184's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for K11n184's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-9 t^2+20 t-25+20 t^{-1} -9 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+3 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 87, 2 }
Jones polynomial q^9-4 q^8+8 q^7-12 q^6+14 q^5-15 q^4+14 q^3-10 q^2+7 q-2
HOMFLY-PT polynomial (db, data sources) 2 z^6 a^{-4} -2 z^4 a^{-2} +8 z^4 a^{-4} -3 z^4 a^{-6} -3 z^2 a^{-2} +11 z^2 a^{-4} -7 z^2 a^{-6} +z^2 a^{-8} +4 a^{-4} -4 a^{-6} + a^{-8}
Kauffman polynomial (db, data sources) 3 z^9 a^{-5} +3 z^9 a^{-7} +8 z^8 a^{-4} +14 z^8 a^{-6} +6 z^8 a^{-8} +6 z^7 a^{-3} +6 z^7 a^{-5} +4 z^7 a^{-7} +4 z^7 a^{-9} +z^6 a^{-2} -20 z^6 a^{-4} -36 z^6 a^{-6} -14 z^6 a^{-8} +z^6 a^{-10} -9 z^5 a^{-3} -27 z^5 a^{-5} -28 z^5 a^{-7} -10 z^5 a^{-9} +8 z^4 a^{-2} +29 z^4 a^{-4} +29 z^4 a^{-6} +6 z^4 a^{-8} -2 z^4 a^{-10} +3 z^3 a^{-1} +11 z^3 a^{-3} +25 z^3 a^{-5} +24 z^3 a^{-7} +7 z^3 a^{-9} -7 z^2 a^{-2} -19 z^2 a^{-4} -14 z^2 a^{-6} -z^2 a^{-8} +z^2 a^{-10} -z a^{-1} -3 z a^{-3} -7 z a^{-5} -7 z a^{-7} -2 z a^{-9} +4 a^{-4} +4 a^{-6} + a^{-8}
The A2 invariant -2+3 q^{-2} - q^{-4} +2 q^{-6} +4 q^{-8} -2 q^{-10} +3 q^{-12} -3 q^{-14} + q^{-16} -3 q^{-20} +2 q^{-22} -2 q^{-24} + q^{-28}
The G2 invariant Data:K11n184/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_84, K11a46,}

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{76}{3} -\frac{28}{3} 128 \frac{160}{3} -\frac{32}{3} -16 \frac{256}{3} 128 \frac{608}{3} -\frac{224}{3} \frac{1111}{15} \frac{436}{15} -\frac{7316}{45} \frac{137}{9} -\frac{329}{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=2 is the signature of K11n184. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-1012345678χ
19         11
17        3 -3
15       51 4
13      73  -4
11     75   2
9    87    -1
7   67     -1
5  48      4
3 36       -3
1 5        5
-12         -2
Integral Khovanov Homology

(db, data source)

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