K11a266

From Knot Atlas
Jump to: navigation, search

K11a265.gif

K11a265

K11a267.gif

K11a267

Contents

K11a266.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a266 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X12,6,13,5 X14,7,15,8 X16,10,17,9 X18,11,19,12 X22,13,1,14 X20,16,21,15 X4,18,5,17 X2,19,3,20 X8,21,9,22
Gauss code 1, -10, 2, -9, 3, -1, 4, -11, 5, -2, 6, -3, 7, -4, 8, -5, 9, -6, 10, -8, 11, -7
Dowker-Thistlethwaite code 6 10 12 14 16 18 22 20 4 2 8
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation K11a266 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a266/ThurstonBennequinNumber
Hyperbolic Volume 20.2286
A-Polynomial See Data:K11a266/A-polynomial

[edit Notes for K11a266's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 0

[edit Notes for K11a266's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-7 t^3+23 t^2-45 t+57-45 t^{-1} +23 t^{-2} -7 t^{-3} + t^{-4}
Conway polynomial z^8+z^6+z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 209, 0 }
Jones polynomial -q^5+5 q^4-12 q^3+21 q^2-29 q+34-34 q^{-1} +30 q^{-2} -22 q^{-3} +14 q^{-4} -6 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) z^8-2 a^2 z^6-z^6 a^{-2} +4 z^6+a^4 z^4-4 a^2 z^4-2 z^4 a^{-2} +6 z^4-z^2 a^{-2} +z^2-a^4+3 a^2+ a^{-2} -2
Kauffman polynomial (db, data sources) 6 a^2 z^{10}+6 z^{10}+15 a^3 z^9+31 a z^9+16 z^9 a^{-1} +14 a^4 z^8+20 a^2 z^8+18 z^8 a^{-2} +24 z^8+6 a^5 z^7-23 a^3 z^7-57 a z^7-16 z^7 a^{-1} +12 z^7 a^{-3} +a^6 z^6-26 a^4 z^6-65 a^2 z^6-26 z^6 a^{-2} +5 z^6 a^{-4} -69 z^6-7 a^5 z^5+2 a^3 z^5+19 a z^5-4 z^5 a^{-1} -13 z^5 a^{-3} +z^5 a^{-5} +10 a^4 z^4+37 a^2 z^4+14 z^4 a^{-2} -3 z^4 a^{-4} +44 z^4+2 a^3 z^3+5 a z^3+7 z^3 a^{-1} +4 z^3 a^{-3} +2 a^4 z^2+a^2 z^2-2 z^2 a^{-2} -3 z^2+a^5 z+a^3 z-a z-z a^{-1} -a^4-3 a^2- a^{-2} -2
The A2 invariant q^{18}-3 q^{16}+3 q^{12}-5 q^{10}+7 q^8-q^6+4 q^2-7+6 q^{-2} -6 q^{-4} +2 q^{-6} +4 q^{-8} -4 q^{-10} +3 q^{-12} - q^{-14}
The G2 invariant Data:K11a266/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, -1)
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 -8 0 0 -8 0 \frac{16}{3} -\frac{32}{3} 24 0 32 0 0 48 \frac{40}{3} \frac{104}{3} -\frac{112}{3} 0

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 K11a266. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
11           1-1
9          4 4
7         81 -7
5        134  9
3       168   -8
1      1813    5
-1     1717     0
-3    1317      -4
-5   917       8
-7  513        -8
-9 19         8
-11 5          -5
-131           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-2 {\mathbb Z}^{17}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=-1 {\mathbb Z}^{17}\oplus{\mathbb Z}_2^{17} {\mathbb Z}^{17}
r=0 {\mathbb Z}^{17}\oplus{\mathbb Z}_2^{17} {\mathbb Z}^{18}
r=1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{16} {\mathbb Z}^{16}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\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).

Back to the top.

K11a265.gif

K11a265

K11a267.gif

K11a267