K11a214

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K11a213.gif

K11a213

K11a215.gif

K11a215

Contents

K11a214.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a214 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X16,5,17,6 X22,8,1,7 X20,10,21,9 X18,12,19,11 X2,13,3,14 X6,15,7,16 X14,18,15,17 X10,20,11,19 X8,22,9,21
Gauss code 1, -7, 2, -1, 3, -8, 4, -11, 5, -10, 6, -2, 7, -9, 8, -3, 9, -6, 10, -5, 11, -4
Dowker-Thistlethwaite code 4 12 16 22 20 18 2 6 14 10 8
A Braid Representative
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A Morse Link Presentation K11a214 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a214/ThurstonBennequinNumber
Hyperbolic Volume 11.7947
A-Polynomial See Data:K11a214/A-polynomial

[edit Notes for K11a214's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant 0

[edit Notes for K11a214's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^2-17 t+29-17 t^{-1} +3 t^{-2}
Conway polynomial 3 z^4-5 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 69, 0 }
Jones polynomial -q^7+3 q^6-4 q^5+7 q^4-9 q^3+10 q^2-11 q+9-7 q^{-1} +5 q^{-2} -2 q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) a^4-2 z^2 a^2+z^4-z^2+z^4 a^{-2} -2 z^2 a^{-2} -2 a^{-2} +z^4 a^{-4} +z^2 a^{-4} +2 a^{-4} -z^2 a^{-6}
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} +3 z^8 a^{-2} +3 z^8 a^{-4} +3 z^8 a^{-6} +3 z^8+3 a z^7-9 z^7 a^{-1} -25 z^7 a^{-3} -12 z^7 a^{-5} +z^7 a^{-7} +3 a^2 z^6-21 z^6 a^{-2} -28 z^6 a^{-4} -14 z^6 a^{-6} -4 z^6+2 a^3 z^5-2 a z^5+17 z^5 a^{-1} +37 z^5 a^{-3} +12 z^5 a^{-5} -4 z^5 a^{-7} +a^4 z^4-3 a^2 z^4+34 z^4 a^{-2} +45 z^4 a^{-4} +18 z^4 a^{-6} +3 z^4-2 a^3 z^3-19 z^3 a^{-1} -29 z^3 a^{-3} -5 z^3 a^{-5} +3 z^3 a^{-7} -2 a^4 z^2+a^2 z^2-21 z^2 a^{-2} -24 z^2 a^{-4} -7 z^2 a^{-6} -z^2+8 z a^{-1} +11 z a^{-3} +3 z a^{-5} +a^4+2 a^{-2} +2 a^{-4}
The A2 invariant Data:K11a214/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a214/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: (-5, -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
-20 -8 200 \frac{746}{3} \frac{238}{3} 160 \frac{592}{3} \frac{64}{3} 24 -\frac{4000}{3} 32 -\frac{14920}{3} -\frac{4760}{3} -\frac{27055}{6} 258 -\frac{27254}{9} \frac{5387}{18} -\frac{3631}{6}

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=0 is the signature of K11a214. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
15           1-1
13          2 2
11         21 -1
9        52  3
7       42   -2
5      65    1
3     54     -1
1    46      -2
-1   46       2
-3  13        -2
-5 14         3
-7 1          -1
-91           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11a213.gif

K11a213

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K11a215