K11a269

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K11a268.gif

K11a268

K11a270.gif

K11a270

Contents

K11a269.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a269 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X12,6,13,5 X20,8,21,7 X16,10,17,9 X18,11,19,12 X22,13,1,14 X8,16,9,15 X4,18,5,17 X2,19,3,20 X14,21,15,22
Gauss code 1, -10, 2, -9, 3, -1, 4, -8, 5, -2, 6, -3, 7, -11, 8, -5, 9, -6, 10, -4, 11, -7
Dowker-Thistlethwaite code 6 10 12 20 16 18 22 8 4 2 14
A Braid Representative
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A Morse Link Presentation K11a269 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a269/ThurstonBennequinNumber
Hyperbolic Volume 18.0173
A-Polynomial See Data:K11a269/A-polynomial

[edit Notes for K11a269's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant -2

[edit Notes for K11a269's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+6 t^3-18 t^2+32 t-37+32 t^{-1} -18 t^{-2} +6 t^{-3} - t^{-4}
Conway polynomial -z^8-2 z^6-2 z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 151, 2 }
Jones polynomial q^7-4 q^6+9 q^5-15 q^4+21 q^3-24 q^2+24 q-21+16 q^{-1} -10 q^{-2} +5 q^{-3} - q^{-4}
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-10 z^4 a^{-2} +3 z^4 a^{-4} +6 z^4-a^2 z^2-8 z^2 a^{-2} +3 z^2 a^{-4} +4 z^2+a^2- a^{-2} + a^{-4}
Kauffman polynomial (db, data sources) 4 z^{10} a^{-2} +4 z^{10}+8 a z^9+20 z^9 a^{-1} +12 z^9 a^{-3} +5 a^2 z^8+15 z^8 a^{-2} +16 z^8 a^{-4} +4 z^8+a^3 z^7-24 a z^7-55 z^7 a^{-1} -16 z^7 a^{-3} +14 z^7 a^{-5} -15 a^2 z^6-63 z^6 a^{-2} -27 z^6 a^{-4} +9 z^6 a^{-6} -42 z^6-2 a^3 z^5+18 a z^5+37 z^5 a^{-1} -5 z^5 a^{-3} -18 z^5 a^{-5} +4 z^5 a^{-7} +12 a^2 z^4+55 z^4 a^{-2} +15 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +44 z^4+a^3 z^3-a z^3-4 z^3 a^{-1} +7 z^3 a^{-3} +8 z^3 a^{-5} -z^3 a^{-7} -a^2 z^2-16 z^2 a^{-2} -5 z^2 a^{-4} +2 z^2 a^{-6} -10 z^2-a z-z a^{-1} -z a^{-3} -z a^{-5} -a^2+ a^{-2} + a^{-4}
The A2 invariant -q^{12}+2 q^{10}+q^8-q^6+4 q^4-4 q^2+1-3 q^{-4} +5 q^{-6} -4 q^{-8} +4 q^{-10} - q^{-12} -2 q^{-14} +3 q^{-16} -2 q^{-18} + q^{-20}
The G2 invariant Data:K11a269/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: (-2, -1)
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
-8 -8 32 \frac{164}{3} \frac{100}{3} 64 \frac{400}{3} \frac{256}{3} -8 -\frac{256}{3} 32 -\frac{1312}{3} -\frac{800}{3} -\frac{4111}{15} \frac{1548}{5} -\frac{29764}{45} \frac{1231}{9} -\frac{2431}{15}

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 K11a269. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123456χ
15           11
13          3 -3
11         61 5
9        93  -6
7       126   6
5      129    -3
3     1212     0
1    1013      3
-1   611       -5
-3  410        6
-5 16         -5
-7 4          4
-91           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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K11a268.gif

K11a268

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K11a270