K11n40

From Knot Atlas
Jump to: navigation, search

K11n39.gif

K11n39

K11n41.gif

K11n41

Contents

K11n40.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n40 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,13,6,12 X2837 X18,10,19,9 X16,11,17,12 X13,7,14,6 X20,15,21,16 X22,18,1,17 X14,19,15,20 X10,22,11,21
Gauss code 1, -4, 2, -1, -3, 7, 4, -2, 5, -11, 6, 3, -7, -10, 8, -6, 9, -5, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 8 -12 2 18 16 -6 20 22 14 10
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11n40 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n40/ThurstonBennequinNumber
Hyperbolic Volume 15.4047
A-Polynomial See Data:K11n40/A-polynomial

[edit Notes for K11n40'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 K11n40's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-8 t^2+18 t-23+18 t^{-1} -8 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+4 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 79, 2 }
Jones polynomial q^9-4 q^8+7 q^7-11 q^6+13 q^5-13 q^4+13 q^3-9 q^2+6 q-2
HOMFLY-PT polynomial (db, data sources) 2 z^6 a^{-4} -2 z^4 a^{-2} +9 z^4 a^{-4} -3 z^4 a^{-6} -4 z^2 a^{-2} +15 z^2 a^{-4} -8 z^2 a^{-6} +z^2 a^{-8} -2 a^{-2} +8 a^{-4} -6 a^{-6} + a^{-8}
Kauffman polynomial (db, data sources) 2 z^9 a^{-5} +2 z^9 a^{-7} +6 z^8 a^{-4} +11 z^8 a^{-6} +5 z^8 a^{-8} +5 z^7 a^{-3} +8 z^7 a^{-5} +7 z^7 a^{-7} +4 z^7 a^{-9} +z^6 a^{-2} -14 z^6 a^{-4} -27 z^6 a^{-6} -11 z^6 a^{-8} +z^6 a^{-10} -9 z^5 a^{-3} -30 z^5 a^{-5} -32 z^5 a^{-7} -11 z^5 a^{-9} +6 z^4 a^{-2} +23 z^4 a^{-4} +22 z^4 a^{-6} +3 z^4 a^{-8} -2 z^4 a^{-10} +3 z^3 a^{-1} +15 z^3 a^{-3} +32 z^3 a^{-5} +28 z^3 a^{-7} +8 z^3 a^{-9} -6 z^2 a^{-2} -20 z^2 a^{-4} -15 z^2 a^{-6} +z^2 a^{-10} -2 z a^{-1} -7 z a^{-3} -13 z a^{-5} -9 z a^{-7} -z a^{-9} +2 a^{-2} +8 a^{-4} +6 a^{-6} + a^{-8}
The A2 invariant -2+2 q^{-2} -2 q^{-4} + q^{-6} +4 q^{-8} +5 q^{-12} - q^{-14} + q^{-16} - q^{-18} -4 q^{-20} + q^{-22} -2 q^{-24} + q^{-28}
The G2 invariant Data:K11n40/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_57, K11n46,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n46,}

Vassiliev invariants

V2 and V3: (4, 6)
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
16 48 128 \frac{680}{3} \frac{88}{3} 768 1152 192 144 \frac{2048}{3} 1152 \frac{10880}{3} \frac{1408}{3} \frac{90422}{15} \frac{1544}{5} \frac{93368}{45} \frac{346}{9} \frac{3782}{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 K11n40. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-1012345678χ
19         11
17        3 -3
15       41 3
13      73  -4
11     64   2
9    77    0
7   66     0
5  37      4
3 36       -3
1 4        4
-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}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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

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.

K11n39.gif

K11n39

K11n41.gif

K11n41