K11a279

From Knot Atlas
Jump to: navigation, search

K11a278.gif

K11a278

K11a280.gif

K11a280

Contents

K11a279.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a279 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X16,5,17,6 X12,8,13,7 X18,9,19,10 X2,11,3,12 X20,14,21,13 X22,16,1,15 X4,17,5,18 X8,19,9,20 X14,22,15,21
Gauss code 1, -6, 2, -9, 3, -1, 4, -10, 5, -2, 6, -4, 7, -11, 8, -3, 9, -5, 10, -7, 11, -8
Dowker-Thistlethwaite code 6 10 16 12 18 2 20 22 4 8 14
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation K11a279 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a279/ThurstonBennequinNumber
Hyperbolic Volume 14.416
A-Polynomial See Data:K11a279/A-polynomial

[edit Notes for K11a279'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 K11a279's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-12 t^2+20 t-21+20 t^{-1} -12 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+6 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 91, 2 }
Jones polynomial -q^6+3 q^5-6 q^4+10 q^3-12 q^2+14 q-14+12 q^{-1} -9 q^{-2} +6 q^{-3} -3 q^{-4} + q^{-5}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +2 z^6-3 a^2 z^4+2 z^4 a^{-2} -z^4 a^{-4} +8 z^4+a^4 z^2-9 a^2 z^2-z^2 a^{-2} -2 z^2 a^{-4} +10 z^2+2 a^4-5 a^2- a^{-2} +5
Kauffman polynomial (db, data sources) 2 a^2 z^{10}+2 z^{10}+3 a^3 z^9+10 a z^9+7 z^9 a^{-1} +a^4 z^8-3 a^2 z^8+11 z^8 a^{-2} +7 z^8-15 a^3 z^7-42 a z^7-15 z^7 a^{-1} +12 z^7 a^{-3} -5 a^4 z^6-18 a^2 z^6-27 z^6 a^{-2} +10 z^6 a^{-4} -50 z^6+24 a^3 z^5+51 a z^5-4 z^5 a^{-1} -25 z^5 a^{-3} +6 z^5 a^{-5} +9 a^4 z^4+44 a^2 z^4+14 z^4 a^{-2} -15 z^4 a^{-4} +3 z^4 a^{-6} +67 z^4-13 a^3 z^3-18 a z^3+11 z^3 a^{-1} +12 z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} -7 a^4 z^2-28 a^2 z^2-4 z^2 a^{-2} +6 z^2 a^{-4} -31 z^2+2 a^3 z+a z-3 z a^{-1} -2 z a^{-3} +2 a^4+5 a^2+ a^{-2} +5
The A2 invariant Data:K11a279/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a279/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: (-1, 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
-4 16 8 -\frac{158}{3} -\frac{130}{3} -64 -\frac{224}{3} -\frac{416}{3} 112 -\frac{32}{3} 128 \frac{632}{3} \frac{520}{3} \frac{14129}{30} -\frac{2818}{15} \frac{30658}{45} -\frac{3377}{18} \frac{3569}{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=2 is the signature of K11a279. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
13           1-1
11          2 2
9         41 -3
7        62  4
5       64   -2
3      86    2
1     77     0
-1    57      -2
-3   47       3
-5  25        -3
-7 14         3
-9 2          -2
-111           1
Integral Khovanov Homology

(db, data source)

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

K11a278.gif

K11a278

K11a280.gif

K11a280