K11n183

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K11n182.gif

K11n182

K11n184.gif

K11n184

Contents

K11n183.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n183 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X3,15,4,14 X10,6,11,5 X7,19,8,18 X2,10,3,9 X22,11,1,12 X20,14,21,13 X15,5,16,4 X12,18,13,17 X19,9,20,8 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.gifBraidPart1.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11n183 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n183/ThurstonBennequinNumber
Hyperbolic Volume 11.4109
A-Polynomial See Data:K11n183/A-polynomial

[edit Notes for K11n183's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -6

[edit Notes for K11n183's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3+t^2-6 t+9-6 t^{-1} + t^{-2} + t^{-3}
Conway polynomial z^6+7 z^4+7 z^2+1
2nd Alexander ideal (db, data sources) \left\{4,t^2+t+1\right\}
Determinant and Signature { 21, 4 }
Jones polynomial q^{12}-3 q^{11}+3 q^{10}-4 q^9+4 q^8-3 q^7+3 q^6-q^5+q^3
HOMFLY-PT polynomial (db, data sources) z^6 a^{-6} +6 z^4 a^{-6} +z^4 a^{-8} +8 z^2 a^{-6} +2 z^2 a^{-8} -3 z^2 a^{-10} +2 a^{-6} +2 a^{-8} -4 a^{-10} + a^{-12}
Kauffman polynomial (db, data sources) 2 z^8 a^{-10} +2 z^8 a^{-12} +2 z^7 a^{-9} +5 z^7 a^{-11} +3 z^7 a^{-13} +z^6 a^{-6} -z^6 a^{-8} -10 z^6 a^{-10} -7 z^6 a^{-12} +z^6 a^{-14} -z^5 a^{-7} -11 z^5 a^{-9} -22 z^5 a^{-11} -12 z^5 a^{-13} -6 z^4 a^{-6} +4 z^4 a^{-8} +17 z^4 a^{-10} +4 z^4 a^{-12} -3 z^4 a^{-14} +2 z^3 a^{-7} +18 z^3 a^{-9} +26 z^3 a^{-11} +10 z^3 a^{-13} +8 z^2 a^{-6} -4 z^2 a^{-8} -14 z^2 a^{-10} -z^2 a^{-12} +z^2 a^{-14} -8 z a^{-9} -9 z a^{-11} -z a^{-13} -2 a^{-6} +2 a^{-8} +4 a^{-10} + a^{-12}
The A2 invariant  q^{-10} + q^{-12} + q^{-16} + q^{-18} +3 q^{-22} + q^{-24} + q^{-26} -2 q^{-28} -3 q^{-30} - q^{-32} -2 q^{-34} + q^{-36} + q^{-38}
The G2 invariant Data:K11n183/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: (7, 17)
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
28 136 392 \frac{2642}{3} \frac{406}{3} 3808 \frac{18928}{3} \frac{3424}{3} 776 \frac{10976}{3} 9248 \frac{73976}{3} \frac{11368}{3} \frac{1387657}{30} \frac{31526}{15} \frac{764234}{45} \frac{2903}{18} \frac{65737}{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=4 is the signature of K11n183. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
012345678910χ
25          11
23         2 -2
21        11 0
19       32  -1
17     121   0
15     23    1
13   132     0
11    2      2
9  12       -1
71          1
51          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5 i=7
r=0 {\mathbb Z} {\mathbb Z}
r=1
r=2 {\mathbb Z}
r=3 {\mathbb Z}^{2} {\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2} {\mathbb Z}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=9 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=10 {\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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K11n182

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K11n184