K11a341

From Knot Atlas
Jump to: navigation, search

K11a340.gif

K11a340

K11a342.gif

K11a342

Contents

K11a341.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a341 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X14,4,15,3 X16,6,17,5 X20,8,21,7 X22,10,1,9 X18,12,19,11 X4,14,5,13 X2,16,3,15 X12,18,13,17 X10,20,11,19 X8,22,9,21
Gauss code 1, -8, 2, -7, 3, -1, 4, -11, 5, -10, 6, -9, 7, -2, 8, -3, 9, -6, 10, -4, 11, -5
Dowker-Thistlethwaite code 6 14 16 20 22 18 4 2 12 10 8
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11a341 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a341/ThurstonBennequinNumber
Hyperbolic Volume 10.4619
A-Polynomial See Data:K11a341/A-polynomial

[edit Notes for K11a341's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant -4

[edit Notes for K11a341's four dimensional invariants]

Polynomial invariants

Alexander polynomial 6 t^2-15 t+19-15 t^{-1} +6 t^{-2}
Conway polynomial 6 z^4+9 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 61, 4 }
Jones polynomial -q^{13}+2 q^{12}-4 q^{11}+6 q^{10}-8 q^9+9 q^8-9 q^7+9 q^6-6 q^5+4 q^4-2 q^3+q^2
HOMFLY-PT polynomial (db, data sources) z^4 a^{-4} +2 z^4 a^{-6} +2 z^4 a^{-8} +z^4 a^{-10} +2 z^2 a^{-4} +4 z^2 a^{-6} +3 z^2 a^{-8} +z^2 a^{-10} -z^2 a^{-12} +2 a^{-6} - a^{-12}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +2 z^9 a^{-9} +4 z^9 a^{-11} +2 z^9 a^{-13} +3 z^8 a^{-8} -z^8 a^{-10} -2 z^8 a^{-12} +2 z^8 a^{-14} +3 z^7 a^{-7} -4 z^7 a^{-9} -16 z^7 a^{-11} -8 z^7 a^{-13} +z^7 a^{-15} +3 z^6 a^{-6} -7 z^6 a^{-8} -z^6 a^{-10} -9 z^6 a^{-14} +2 z^5 a^{-5} -5 z^5 a^{-7} +4 z^5 a^{-9} +26 z^5 a^{-11} +10 z^5 a^{-13} -5 z^5 a^{-15} +z^4 a^{-4} -6 z^4 a^{-6} +8 z^4 a^{-8} +z^4 a^{-10} -3 z^4 a^{-12} +11 z^4 a^{-14} -3 z^3 a^{-5} +4 z^3 a^{-7} -2 z^3 a^{-9} -24 z^3 a^{-11} -8 z^3 a^{-13} +7 z^3 a^{-15} -2 z^2 a^{-4} +6 z^2 a^{-6} -2 z^2 a^{-10} +3 z^2 a^{-12} -3 z^2 a^{-14} +z a^{-7} +z a^{-9} +6 z a^{-11} +4 z a^{-13} -2 z a^{-15} -2 a^{-6} - a^{-12}
The A2 invariant Data:K11a341/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a341/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, 27)
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
36 216 648 1674 270 7776 14288 2464 2104 7776 23328 60264 9720 \frac{1235773}{10} \frac{2174}{5} \frac{259382}{5} \frac{2305}{2} \frac{70173}{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=4 is the signature of K11a341. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
27           1-1
25          1 1
23         31 -2
21        31  2
19       53   -2
17      43    1
15     55     0
13    44      0
11   25       3
9  24        -2
7  2         2
512          -1
31           1
Integral Khovanov Homology

(db, data source)

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

K11a340.gif

K11a340

K11a342.gif

K11a342