K11a334

From Knot Atlas
Jump to: navigation, search

K11a333.gif

K11a333

K11a335.gif

K11a335

Contents

K11a334.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a334 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X14,4,15,3 X16,6,17,5 X18,8,19,7 X20,10,21,9 X22,12,1,11 X2,14,3,13 X4,16,5,15 X12,18,13,17 X8,20,9,19 X10,22,11,21
Gauss code 1, -7, 2, -8, 3, -1, 4, -10, 5, -11, 6, -9, 7, -2, 8, -3, 9, -4, 10, -5, 11, -6
Dowker-Thistlethwaite code 6 14 16 18 20 22 2 4 12 8 10
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11a334 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 4
3-genus 4
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a334/ThurstonBennequinNumber
Hyperbolic Volume 8.78087
A-Polynomial See Data:K11a334/A-polynomial

[edit Notes for K11a334'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 K11a334's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^4-4 t^3+6 t^2-8 t+9-8 t^{-1} +6 t^{-2} -4 t^{-3} +2 t^{-4}
Conway polynomial 2 z^8+12 z^6+22 z^4+12 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 49, 8 }
Jones polynomial -q^{15}+2 q^{14}-4 q^{13}+6 q^{12}-7 q^{11}+7 q^{10}-7 q^9+6 q^8-4 q^7+3 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} +6 z^6 a^{-10} -z^6 a^{-12} +16 z^4 a^{-8} +11 z^4 a^{-10} -5 z^4 a^{-12} +13 z^2 a^{-8} +6 z^2 a^{-10} -7 z^2 a^{-12} +3 a^{-8} -2 a^{-12}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +z^9 a^{-9} +3 z^9 a^{-11} +2 z^9 a^{-13} +z^8 a^{-8} -5 z^8 a^{-10} -3 z^8 a^{-12} +3 z^8 a^{-14} -5 z^7 a^{-9} -14 z^7 a^{-11} -5 z^7 a^{-13} +4 z^7 a^{-15} -7 z^6 a^{-8} +8 z^6 a^{-10} +5 z^6 a^{-12} -6 z^6 a^{-14} +4 z^6 a^{-16} +6 z^5 a^{-9} +20 z^5 a^{-11} +3 z^5 a^{-13} -8 z^5 a^{-15} +3 z^5 a^{-17} +16 z^4 a^{-8} -6 z^4 a^{-10} -13 z^4 a^{-12} +z^4 a^{-14} -6 z^4 a^{-16} +2 z^4 a^{-18} +z^3 a^{-9} -11 z^3 a^{-11} -4 z^3 a^{-13} +5 z^3 a^{-15} -2 z^3 a^{-17} +z^3 a^{-19} -13 z^2 a^{-8} +3 z^2 a^{-10} +11 z^2 a^{-12} +4 z^2 a^{-16} -z^2 a^{-18} -2 z a^{-9} +2 z a^{-11} +z a^{-13} -z a^{-15} +z a^{-17} -z a^{-19} +3 a^{-8} -2 a^{-12}
The A2 invariant  q^{-14} +2 q^{-18} +2 q^{-22} + q^{-24} + q^{-28} -2 q^{-30} + q^{-32} - q^{-34} - q^{-40} - q^{-44}
The G2 invariant Data:K11a334/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: (12, 40)
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
48 320 1152 2760 392 15360 \frac{79424}{3} \frac{13856}{3} 3168 18432 51200 132480 18816 \frac{1300102}{5} \frac{177256}{15} \frac{465176}{5} 1534 \frac{58182}{5}

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 K11a334. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
31           1-1
29          1 1
27         31 -2
25        31  2
23       43   -1
21      33    0
19     44     0
17    23      -1
15   24       2
13  12        -1
11  2         2
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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=8 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=9 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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.

K11a333.gif

K11a333

K11a335.gif

K11a335