K11n168

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K11n167.gif

K11n167

K11n169.gif

K11n169

Contents

K11n168.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n168 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X14,6,15,5 X20,8,21,7 X4,10,5,9 X11,18,12,19 X2,14,3,13 X22,15,1,16 X8,18,9,17 X19,12,20,13 X16,21,17,22
Gauss code 1, -7, 2, -5, 3, -1, 4, -9, 5, -2, -6, 10, 7, -3, 8, -11, 9, 6, -10, -4, 11, -8
Dowker-Thistlethwaite code 6 10 14 20 4 -18 2 22 8 -12 16
A Braid Representative
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A Morse Link Presentation K11n168 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:K11n168/ThurstonBennequinNumber
Hyperbolic Volume 15.0132
A-Polynomial See Data:K11n168/A-polynomial

[edit Notes for K11n168'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 K11n168's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-6 t^2+18 t-25+18 t^{-1} -6 t^{-2} + t^{-3}
Conway polynomial z^6+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 75, 2 }
Jones polynomial -2 q^6+5 q^5-9 q^4+12 q^3-12 q^2+13 q-10+7 q^{-1} -4 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +3 z^4 a^{-2} -z^4 a^{-4} -2 z^4+a^2 z^2+5 z^2 a^{-2} -3 z^2+3 a^{-2} - a^{-6} -1
Kauffman polynomial (db, data sources) 2 z^9 a^{-1} +2 z^9 a^{-3} +9 z^8 a^{-2} +4 z^8 a^{-4} +5 z^8+4 a z^7+4 z^7 a^{-1} +3 z^7 a^{-3} +3 z^7 a^{-5} +a^2 z^6-18 z^6 a^{-2} -4 z^6 a^{-4} +z^6 a^{-6} -12 z^6-11 a z^5-18 z^5 a^{-1} -7 z^5 a^{-3} -2 a^2 z^4+6 z^4 a^{-2} +3 z^4 a^{-4} +4 z^4 a^{-6} +5 z^4+7 a z^3+6 z^3 a^{-1} -3 z^3 a^{-3} +z^3 a^{-5} +3 z^3 a^{-7} +a^2 z^2+z^2 a^{-2} -z^2 a^{-4} -2 z^2 a^{-6} +z^2+a z+3 z a^{-1} +4 z a^{-3} -2 z a^{-7} -3 a^{-2} + a^{-6} -1
The A2 invariant Data:K11n168/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n168/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n148,}

Vassiliev invariants

V2 and V3: (3, 4)
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
12 32 72 158 34 384 \frac{2144}{3} \frac{224}{3} 192 288 512 1896 408 \frac{33871}{10} -\frac{2302}{5} \frac{30302}{15} \frac{305}{6} \frac{3471}{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=2 is the signature of K11n168. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-1012345χ
13         2-2
11        3 3
9       62 -4
7      63  3
5     66   0
3    76    1
1   47     3
-1  36      -3
-3 14       3
-5 3        -3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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).

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K11n167.gif

K11n167

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K11n169