K11a240

From Knot Atlas
Jump to: navigation, search

K11a239.gif

K11a239

K11a241.gif

K11a241

Contents

K11a240.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a240 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X14,4,15,3 X18,6,19,5 X20,8,21,7 X22,10,1,9 X16,12,17,11 X2,14,3,13 X10,16,11,15 X12,18,13,17 X6,20,7,19 X8,22,9,21
Gauss code 1, -7, 2, -1, 3, -10, 4, -11, 5, -8, 6, -9, 7, -2, 8, -6, 9, -3, 10, -4, 11, -5
Dowker-Thistlethwaite code 4 14 18 20 22 16 2 10 12 6 8
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11a240 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 4
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a240/ThurstonBennequinNumber
Hyperbolic Volume 10.6981
A-Polynomial See Data:K11a240/A-polynomial

[edit Notes for K11a240's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant -8

[edit Notes for K11a240's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^4-5 t^3+8 t^2-10 t+11-10 t^{-1} +8 t^{-2} -5 t^{-3} +2 t^{-4}
Conway polynomial 2 z^8+11 z^6+18 z^4+9 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 61, 8 }
Jones polynomial -q^{15}+3 q^{14}-5 q^{13}+7 q^{12}-9 q^{11}+9 q^{10}-9 q^9+7 q^8-5 q^7+4 q^6-q^5+q^4
HOMFLY-PT polynomial (db, data sources) z^8 a^{-8} +z^8 a^{-10} +7 z^6 a^{-8} +5 z^6 a^{-10} -z^6 a^{-12} +17 z^4 a^{-8} +5 z^4 a^{-10} -4 z^4 a^{-12} +17 z^2 a^{-8} -5 z^2 a^{-10} -3 z^2 a^{-12} +6 a^{-8} -6 a^{-10} + a^{-12}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +z^9 a^{-9} +4 z^9 a^{-11} +3 z^9 a^{-13} +z^8 a^{-8} -3 z^8 a^{-10} +z^8 a^{-12} +5 z^8 a^{-14} -4 z^7 a^{-9} -16 z^7 a^{-11} -6 z^7 a^{-13} +6 z^7 a^{-15} -7 z^6 a^{-8} -3 z^6 a^{-10} -13 z^6 a^{-12} -11 z^6 a^{-14} +6 z^6 a^{-16} +z^5 a^{-9} +13 z^5 a^{-11} -3 z^5 a^{-13} -10 z^5 a^{-15} +5 z^5 a^{-17} +17 z^4 a^{-8} +15 z^4 a^{-10} +12 z^4 a^{-12} +4 z^4 a^{-14} -7 z^4 a^{-16} +3 z^4 a^{-18} +9 z^3 a^{-9} +7 z^3 a^{-11} +5 z^3 a^{-13} +2 z^3 a^{-15} -4 z^3 a^{-17} +z^3 a^{-19} -17 z^2 a^{-8} -15 z^2 a^{-10} -z^2 a^{-12} -z^2 a^{-14} +z^2 a^{-16} -z^2 a^{-18} -7 z a^{-9} -7 z a^{-11} -z a^{-13} +z a^{-17} +6 a^{-8} +6 a^{-10} + a^{-12}
The A2 invariant  q^{-14} +3 q^{-18} + q^{-20} +3 q^{-22} + q^{-24} - q^{-26} -4 q^{-30} -2 q^{-34} + q^{-38} + q^{-42} - q^{-44}
The G2 invariant Data:K11a240/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: (9, 25)
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
36 200 648 1434 174 7200 \frac{34640}{3} \frac{5600}{3} 1192 7776 20000 51624 6264 \frac{962893}{10} \frac{30094}{5} \frac{150022}{5} \frac{1361}{2} \frac{35053}{10}

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=8 is the signature of K11a240. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
31           1-1
29          2 2
27         31 -2
25        42  2
23       53   -2
21      44    0
19     55     0
17    24      -2
15   35       2
13  12        -1
11  3         3
911          0
71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=7 i=9
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=8 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=9 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=11 {\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.

K11a239.gif

K11a239

K11a241.gif

K11a241