K11n146

From Knot Atlas
Jump to: navigation, search

K11n145.gif

K11n145

K11n147.gif

K11n147

Contents

K11n146.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n146 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X14,4,15,3 X10,5,11,6 X20,8,21,7 X22,9,1,10 X11,19,12,18 X2,14,3,13 X8,15,9,16 X6,18,7,17 X19,13,20,12 X16,22,17,21
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.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11n146 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n146/ThurstonBennequinNumber
Hyperbolic Volume 14.497
A-Polynomial See Data:K11n146/A-polynomial

[edit Notes for K11n146'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 K11n146's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-5 t^2+15 t-21+15 t^{-1} -5 t^{-2} + t^{-3}
Conway polynomial z^6+z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 63, 2 }
Jones polynomial q^9-4 q^8+6 q^7-9 q^6+11 q^5-10 q^4+10 q^3-7 q^2+4 q-1
HOMFLY-PT polynomial (db, data sources) z^6 a^{-4} -z^4 a^{-2} +4 z^4 a^{-4} -2 z^4 a^{-6} -z^2 a^{-2} +8 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} - a^{-2} +5 a^{-4} -3 a^{-6}
Kauffman polynomial (db, data sources) 2 z^9 a^{-5} +2 z^9 a^{-7} +4 z^8 a^{-4} +9 z^8 a^{-6} +5 z^8 a^{-8} +2 z^7 a^{-3} +2 z^7 a^{-7} +4 z^7 a^{-9} -13 z^6 a^{-4} -29 z^6 a^{-6} -15 z^6 a^{-8} +z^6 a^{-10} -3 z^5 a^{-3} -11 z^5 a^{-5} -20 z^5 a^{-7} -12 z^5 a^{-9} +4 z^4 a^{-2} +24 z^4 a^{-4} +33 z^4 a^{-6} +11 z^4 a^{-8} -2 z^4 a^{-10} +z^3 a^{-1} +8 z^3 a^{-3} +18 z^3 a^{-5} +18 z^3 a^{-7} +7 z^3 a^{-9} -3 z^2 a^{-2} -17 z^2 a^{-4} -18 z^2 a^{-6} -4 z^2 a^{-8} -z a^{-1} -4 z a^{-3} -8 z a^{-5} -4 z a^{-7} +z a^{-9} + a^{-2} +5 a^{-4} +3 a^{-6}
The A2 invariant -1+2 q^{-2} -2 q^{-4} +3 q^{-8} +4 q^{-12} + q^{-16} -3 q^{-20} + q^{-22} -2 q^{-24} - q^{-26} + q^{-28}
The G2 invariant Data:K11n146/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n167,}

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

Vassiliev invariants

V2 and V3: (4, 7)
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 56 128 \frac{872}{3} \frac{160}{3} 896 \frac{4784}{3} \frac{896}{3} 248 \frac{2048}{3} 1568 \frac{13952}{3} \frac{2560}{3} \frac{132422}{15} -\frac{688}{15} \frac{179408}{45} \frac{346}{9} \frac{8822}{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 K11n146. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-1012345678χ
19         11
17        3 -3
15       31 2
13      63  -3
11     53   2
9    56    1
7   55     0
5  25      3
3 25       -3
1 3        3
-11         -1
Integral Khovanov Homology

(db, data source)

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

K11n145.gif

K11n145

K11n147.gif

K11n147