K11n164

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K11n163.gif

K11n163

K11n165.gif

K11n165

Contents

K11n164.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n164 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X3,11,4,10 X14,5,15,6 X16,8,17,7 X9,21,10,20 X11,5,12,4 X13,19,14,18 X2,15,3,16 X22,18,1,17 X19,13,20,12 X21,9,22,8
Gauss code 1, -8, -2, 6, 3, -1, 4, 11, -5, 2, -6, 10, -7, -3, 8, -4, 9, 7, -10, 5, -11, -9
Dowker-Thistlethwaite code 6 -10 14 16 -20 -4 -18 2 22 -12 -8
A Braid Representative
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A Morse Link Presentation K11n164 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n164/ThurstonBennequinNumber
Hyperbolic Volume 13.1502
A-Polynomial See Data:K11n164/A-polynomial

[edit Notes for K11n164's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -4

[edit Notes for K11n164's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+5 t^2-10 t+13-10 t^{-1} +5 t^{-2} - t^{-3}
Conway polynomial -z^6-z^4+z^2+1
2nd Alexander ideal (db, data sources) \left\{t^2-t+1\right\}
Determinant and Signature { 45, 4 }
Jones polynomial 2 q^8-5 q^7+6 q^6-8 q^5+8 q^4-6 q^3+6 q^2-3 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} +z^4 a^{-2} -3 z^4 a^{-4} +z^4 a^{-6} +2 z^2 a^{-2} -z^2 a^{-4} + a^{-2} +2 a^{-4} -3 a^{-6} + a^{-8}
Kauffman polynomial (db, data sources) z^9 a^{-5} +z^9 a^{-7} +3 z^8 a^{-4} +4 z^8 a^{-6} +z^8 a^{-8} +3 z^7 a^{-3} +2 z^7 a^{-5} -z^7 a^{-7} +z^6 a^{-2} -7 z^6 a^{-4} -10 z^6 a^{-6} -2 z^6 a^{-8} -9 z^5 a^{-3} -12 z^5 a^{-5} -2 z^5 a^{-7} +z^5 a^{-9} -3 z^4 a^{-2} -z^4 a^{-4} +6 z^4 a^{-6} +4 z^4 a^{-8} +5 z^3 a^{-3} +9 z^3 a^{-5} +8 z^3 a^{-7} +4 z^3 a^{-9} +3 z^2 a^{-2} +2 z^2 a^{-4} -3 z^2 a^{-6} +2 z^2 a^{-10} -4 z a^{-5} -7 z a^{-7} -3 z a^{-9} - a^{-2} +2 a^{-4} +3 a^{-6} + a^{-8}
The A2 invariant Data:K11n164/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n164/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_18, 9_24, K11n85,}

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

Vassiliev invariants

V2 and V3: (1, 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
4 0 8 -\frac{34}{3} \frac{34}{3} 0 -32 0 64 \frac{32}{3} 0 -\frac{136}{3} \frac{136}{3} -\frac{1649}{30} -\frac{4862}{15} \frac{21662}{45} \frac{1265}{18} \frac{2191}{30}

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 K11n164. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456χ
17        22
15       3 -3
13      32 1
11     53  -2
9    33   0
7   35    2
5  33     0
3 14      3
1 2       -2
-11        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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K11n163.gif

K11n163

K11n165.gif

K11n165