K11n145

From Knot Atlas
Jump to: navigation, search

K11n144.gif

K11n144

K11n146.gif

K11n146

Contents

K11n145.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n145 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X14,4,15,3 X5,11,6,10 X7,20,8,21 X9,1,10,22 X18,11,19,12 X2,14,3,13 X15,9,16,8 X17,6,18,7 X12,19,13,20 X21,16,22,17
Gauss code 1, -7, 2, -1, -3, 9, -4, 8, -5, 3, 6, -10, 7, -2, -8, 11, -9, -6, 10, 4, -11, 5
Dowker-Thistlethwaite code 4 14 -10 -20 -22 18 2 -8 -6 12 -16
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation K11n145 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n145/ThurstonBennequinNumber
Hyperbolic Volume 9.01207
A-Polynomial See Data:K11n145/A-polynomial

[edit Notes for K11n145's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [0,3]
Rasmussen s-Invariant 0

[edit Notes for K11n145's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-t^2-3 t+7-3 t^{-1} - t^{-2} + t^{-3}
Conway polynomial z^6+5 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 9, 0 }
Jones polynomial -q^5+2 q^4-2 q^3+2 q^2-2 q+2+ q^{-3} - q^{-4}
HOMFLY-PT polynomial (db, data sources) z^6-a^2 z^4+6 z^4-4 a^2 z^2-2 z^2 a^{-2} -z^2 a^{-4} +9 z^2-2 a^2-2 a^{-2} +5
Kauffman polynomial (db, data sources) a^2 z^8+z^8+a^3 z^7+a z^7+z^7 a^{-1} +z^7 a^{-3} -7 a^2 z^6+z^6 a^{-2} +2 z^6 a^{-4} -8 z^6-6 a^3 z^5-8 a z^5-6 z^5 a^{-1} -3 z^5 a^{-3} +z^5 a^{-5} +14 a^2 z^4-2 z^4 a^{-2} -7 z^4 a^{-4} +19 z^4+9 a^3 z^3+15 a z^3+10 z^3 a^{-1} +z^3 a^{-3} -3 z^3 a^{-5} -10 a^2 z^2-2 z^2 a^{-2} +4 z^2 a^{-4} -16 z^2-3 a^3 z-6 a z-4 z a^{-1} +z a^{-5} +2 a^2+2 a^{-2} +5
The A2 invariant -q^{12}-q^{10}+q^6+2 q^2+2+ q^{-2} + q^{-4} - q^{-6} - q^{-10} + q^{-14} - q^{-16}
The G2 invariant Data:K11n145/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: (2, 0)
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
8 0 32 \frac{28}{3} -\frac{76}{3} 0 -32 0 -32 \frac{256}{3} 0 \frac{224}{3} -\frac{608}{3} \frac{511}{15} \frac{1132}{5} -\frac{19316}{45} \frac{401}{9} -\frac{2129}{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=0 is the signature of K11n145. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-1012345χ
11          1-1
9         1 1
7        11 0
5      121  0
3      11   0
1    132    0
-1   112     2
-3   11      0
-5 111       1
-7           0
-91          -1
Integral Khovanov Homology

(db, data source)

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

K11n144.gif

K11n144

K11n146.gif

K11n146