K11n23

From Knot Atlas
Jump to: navigation, search

K11n22.gif

K11n22

K11n24.gif

K11n24

Contents

K11n23.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n23 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,13,6,12 X2837 X9,15,10,14 X11,19,12,18 X13,7,14,6 X15,20,16,21 X17,11,18,10 X19,22,20,1 X21,16,22,17
Gauss code 1, -4, 2, -1, -3, 7, 4, -2, -5, 9, -6, 3, -7, 5, -8, 11, -9, 6, -10, 8, -11, 10
Dowker-Thistlethwaite code 4 8 -12 2 -14 -18 -6 -20 -10 -22 -16
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11n23 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{2,3\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n23/ThurstonBennequinNumber
Hyperbolic Volume 10.8083
A-Polynomial See Data:K11n23/A-polynomial

[edit Notes for K11n23's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n23's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-3 t^3+5 t^2-4 t+3-4 t^{-1} +5 t^{-2} -3 t^{-3} + t^{-4}
Conway polynomial z^8+5 z^6+7 z^4+5 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 29, 4 }
Jones polynomial -2 q^7+3 q^6-4 q^5+5 q^4-4 q^3+5 q^2-3 q+2- q^{-1}
HOMFLY-PT polynomial (db, data sources) z^8 a^{-4} -z^6 a^{-2} +7 z^6 a^{-4} -z^6 a^{-6} -5 z^4 a^{-2} +18 z^4 a^{-4} -6 z^4 a^{-6} -7 z^2 a^{-2} +22 z^2 a^{-4} -11 z^2 a^{-6} +z^2 a^{-8} -3 a^{-2} +10 a^{-4} -7 a^{-6} + a^{-8}
Kauffman polynomial (db, data sources) z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +5 z^8 a^{-4} +3 z^8 a^{-6} +z^7 a^{-1} -z^7 a^{-3} +2 z^7 a^{-7} -10 z^6 a^{-2} -25 z^6 a^{-4} -15 z^6 a^{-6} -5 z^5 a^{-1} -12 z^5 a^{-3} -16 z^5 a^{-5} -9 z^5 a^{-7} +15 z^4 a^{-2} +39 z^4 a^{-4} +25 z^4 a^{-6} +z^4 a^{-8} +7 z^3 a^{-1} +21 z^3 a^{-3} +27 z^3 a^{-5} +13 z^3 a^{-7} -10 z^2 a^{-2} -29 z^2 a^{-4} -20 z^2 a^{-6} -z^2 a^{-8} -3 z a^{-1} -9 z a^{-3} -13 z a^{-5} -6 z a^{-7} +z a^{-9} +3 a^{-2} +10 a^{-4} +7 a^{-6} + a^{-8}
The A2 invariant -q^2- q^{-2} + q^{-6} +2 q^{-8} +4 q^{-10} + q^{-12} +3 q^{-14} - q^{-16} - q^{-18} -2 q^{-20} -2 q^{-22} - q^{-26} + q^{-28}
The G2 invariant Data:K11n23/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: (5, 8)
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
20 64 200 \frac{1030}{3} \frac{122}{3} 1280 \frac{5728}{3} \frac{928}{3} 224 \frac{4000}{3} 2048 \frac{20600}{3} \frac{2440}{3} \frac{67135}{6} \frac{1666}{3} \frac{33662}{9} \frac{1669}{18} \frac{2623}{6}

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 K11n23. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345χ
15        2-2
13       1 1
11      32 -1
9     21  1
7    23   1
5   32    1
3  13     2
1 12      -1
-1 1       1
-31        -1
Integral Khovanov Homology

(db, data source)

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

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.

K11n22.gif

K11n22

K11n24.gif

K11n24