K11n169

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K11n168.gif

K11n168

K11n170.gif

K11n170

Contents

K11n169.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n169 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X3,11,4,10 X5,17,6,16 X12,8,13,7 X9,19,10,18 X11,3,12,2 X20,14,21,13 X22,16,1,15 X17,5,18,4 X19,9,20,8 X14,22,15,21
Gauss code 1, 6, -2, 9, -3, -1, 4, 10, -5, 2, -6, -4, 7, -11, 8, 3, -9, 5, -10, -7, 11, -8
Dowker-Thistlethwaite code 6 -10 -16 12 -18 -2 20 22 -4 -8 14
A Braid Representative
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A Morse Link Presentation K11n169 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{3,4\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n169/ThurstonBennequinNumber
Hyperbolic Volume 10.8173
A-Polynomial See Data:K11n169/A-polynomial

[edit Notes for K11n169's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -6

[edit Notes for K11n169's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-6 t^2+6 t-5+6 t^{-1} -6 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+12 z^4+9 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 35, 6 }
Jones polynomial -q^{12}+2 q^{11}-4 q^{10}+5 q^9-6 q^8+6 q^7-4 q^6+4 q^5-2 q^4+q^3
HOMFLY-PT polynomial (db, data sources) z^6 a^{-6} +2 z^6 a^{-8} +4 z^4 a^{-6} +10 z^4 a^{-8} -2 z^4 a^{-10} +3 z^2 a^{-6} +14 z^2 a^{-8} -8 z^2 a^{-10} +6 a^{-8} -6 a^{-10} + a^{-12}
Kauffman polynomial (db, data sources) z^9 a^{-9} +z^9 a^{-11} +3 z^8 a^{-8} +4 z^8 a^{-10} +z^8 a^{-12} +2 z^7 a^{-7} -z^7 a^{-9} -3 z^7 a^{-11} +z^6 a^{-6} -14 z^6 a^{-8} -18 z^6 a^{-10} -3 z^6 a^{-12} -7 z^5 a^{-7} -7 z^5 a^{-9} +3 z^5 a^{-11} +3 z^5 a^{-13} -4 z^4 a^{-6} +22 z^4 a^{-8} +30 z^4 a^{-10} +6 z^4 a^{-12} +2 z^4 a^{-14} +4 z^3 a^{-7} +11 z^3 a^{-9} +2 z^3 a^{-11} -4 z^3 a^{-13} +z^3 a^{-15} +3 z^2 a^{-6} -18 z^2 a^{-8} -22 z^2 a^{-10} -3 z^2 a^{-12} -2 z^2 a^{-14} -5 z a^{-9} -2 z a^{-11} +2 z a^{-13} -z a^{-15} +6 a^{-8} +6 a^{-10} + a^{-12}
The A2 invariant Data:K11n169/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n169/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: (9, 25)
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
36 200 648 1482 222 7200 \frac{36464}{3} \frac{6368}{3} 1544 7776 20000 53352 7992 \frac{1019133}{10} \frac{17854}{5} \frac{190502}{5} \frac{1313}{2} \frac{49213}{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=6 is the signature of K11n169. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
0123456789χ
25         1-1
23        1 1
21       31 -2
19      21  1
17     43   -1
15    22    0
13   24     2
11  22      0
9  2       2
712        -1
51         1
Integral Khovanov Homology

(db, data source)

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

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K11n168.gif

K11n168

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K11n170