K11a159

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K11a158.gif

K11a158

K11a160.gif

K11a160

Contents

K11a159.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a159 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X18,6,19,5 X14,8,15,7 X2,10,3,9 X22,11,1,12 X8,14,9,13 X20,15,21,16 X6,18,7,17 X16,19,17,20 X12,21,13,22
Gauss code 1, -5, 2, -1, 3, -9, 4, -7, 5, -2, 6, -11, 7, -4, 8, -10, 9, -3, 10, -8, 11, -6
Dowker-Thistlethwaite code 4 10 18 14 2 22 8 20 6 16 12
A Braid Representative
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A Morse Link Presentation K11a159 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a159/ThurstonBennequinNumber
Hyperbolic Volume 14.4089
A-Polynomial See Data:K11a159/A-polynomial

[edit Notes for K11a159'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 K11a159's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-9 t^2+27 t-37+27 t^{-1} -9 t^{-2} + t^{-3}
Conway polynomial z^6-3 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 111, 2 }
Jones polynomial -q^8+3 q^7-7 q^6+12 q^5-15 q^4+18 q^3-18 q^2+15 q-11+7 q^{-1} -3 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +2 z^4 a^{-2} -3 z^4 a^{-4} -2 z^4+a^2 z^2+3 z^2 a^{-2} -4 z^2 a^{-4} +3 z^2 a^{-6} -3 z^2+a^2+ a^{-2} - a^{-4} +2 a^{-6} - a^{-8} -1
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +6 z^9 a^{-3} +3 z^9 a^{-5} +7 z^8 a^{-2} +8 z^8 a^{-4} +5 z^8 a^{-6} +4 z^8+3 a z^7-3 z^7 a^{-3} +5 z^7 a^{-5} +5 z^7 a^{-7} +a^2 z^6-13 z^6 a^{-2} -11 z^6 a^{-4} -3 z^6 a^{-6} +3 z^6 a^{-8} -7 z^6-8 a z^5-10 z^5 a^{-1} -8 z^5 a^{-3} -14 z^5 a^{-5} -7 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+z^4 a^{-2} -4 z^4 a^{-6} -5 z^4 a^{-8} -z^4+6 a z^3+6 z^3 a^{-1} +4 z^3 a^{-3} +9 z^3 a^{-5} +3 z^3 a^{-7} -2 z^3 a^{-9} +3 a^2 z^2+3 z^2 a^{-2} +3 z^2 a^{-4} +5 z^2 a^{-6} +3 z^2 a^{-8} +5 z^2-a z+z a^{-3} -z a^{-5} +z a^{-9} -a^2- a^{-2} - a^{-4} -2 a^{-6} - a^{-8} -1
The A2 invariant Data:K11a159/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a159/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a347,}

Vassiliev invariants

V2 and V3: (0, 3)
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
0 24 0 96 24 0 272 0 120 0 288 0 0 1088 -408 856 112 128

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 K11a159. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          2 2
13         51 -4
11        72  5
9       85   -3
7      107    3
5     88     0
3    710      -3
1   59       4
-1  26        -4
-3 15         4
-5 2          -2
-71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\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).

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K11a158.gif

K11a158

K11a160.gif

K11a160