K11n149

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K11n148.gif

K11n148

K11n150.gif

K11n150

Contents

K11n149.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n149 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X8394 X10,6,11,5 X18,8,19,7 X16,9,17,10 X11,21,12,20 X13,1,14,22 X4,16,5,15 X2,17,3,18 X19,15,20,14 X21,13,22,12
Gauss code 1, -9, 2, -8, 3, -1, 4, -2, 5, -3, -6, 11, -7, 10, 8, -5, 9, -4, -10, 6, -11, 7
Dowker-Thistlethwaite code 6 8 10 18 16 -20 -22 4 2 -14 -12
A Braid Representative
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A Morse Link Presentation K11n149 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n149/ThurstonBennequinNumber
Hyperbolic Volume 11.1363
A-Polynomial See Data:K11n149/A-polynomial

[edit Notes for K11n149's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n149's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-4 t^3+6 t^2-4 t+3-4 t^{-1} +6 t^{-2} -4 t^{-3} + t^{-4}
Conway polynomial z^8+4 z^6+2 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 33, 4 }
Jones polynomial q^8-2 q^7+3 q^6-5 q^5+5 q^4-5 q^3+5 q^2-3 q+3- q^{-1}
HOMFLY-PT polynomial (db, data sources) z^8 a^{-4} -z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -4 z^4 a^{-2} +11 z^4 a^{-4} -5 z^4 a^{-6} -2 z^2 a^{-2} +7 z^2 a^{-4} -6 z^2 a^{-6} +z^2 a^{-8} +2 a^{-2} -2 a^{-6} + a^{-8}
Kauffman polynomial (db, data sources) 2 z^9 a^{-3} +2 z^9 a^{-5} +3 z^8 a^{-2} +6 z^8 a^{-4} +3 z^8 a^{-6} +z^7 a^{-1} -8 z^7 a^{-3} -8 z^7 a^{-5} +z^7 a^{-7} -15 z^6 a^{-2} -29 z^6 a^{-4} -14 z^6 a^{-6} -4 z^5 a^{-1} +6 z^5 a^{-3} +9 z^5 a^{-5} -z^5 a^{-7} +20 z^4 a^{-2} +39 z^4 a^{-4} +22 z^4 a^{-6} +3 z^4 a^{-8} +3 z^3 a^{-1} -z^3 a^{-3} -8 z^3 a^{-5} -2 z^3 a^{-7} +2 z^3 a^{-9} -6 z^2 a^{-2} -16 z^2 a^{-4} -15 z^2 a^{-6} -4 z^2 a^{-8} +z^2 a^{-10} +3 z a^{-3} +5 z a^{-5} +z a^{-7} -z a^{-9} -2 a^{-2} +2 a^{-6} + a^{-8}
The A2 invariant -q^2+1+ q^{-2} +2 q^{-4} +2 q^{-6} + q^{-8} + q^{-10} -3 q^{-12} -2 q^{-16} + q^{-24} - q^{-26} + q^{-28}
The G2 invariant Data:K11n149/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, -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
0 -16 0 -80 -16 0 -\frac{640}{3} -\frac{64}{3} -48 0 128 0 0 -136 -48 \frac{352}{3} -\frac{296}{3} 40

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 K11n149. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-10123456χ
17         11
15        1 -1
13       21 1
11      31  -2
9     22   0
7    33    0
5   22     0
3  24      2
1 11       0
-1 2        2
-31         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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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K11n148.gif

K11n148

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K11n150