K11a16

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K11a15.gif

K11a15

K11a17.gif

K11a17

Contents

K11a16.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a16 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X12,5,13,6 X2837 X14,9,15,10 X18,12,19,11 X6,13,7,14 X22,15,1,16 X20,17,21,18 X10,20,11,19 X16,21,17,22
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -3, 7, -5, 8, -11, 9, -6, 10, -9, 11, -8
Dowker-Thistlethwaite code 4 8 12 2 14 18 6 22 20 10 16
A Braid Representative
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A Morse Link Presentation K11a16 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:K11a16/ThurstonBennequinNumber
Hyperbolic Volume 14.4579
A-Polynomial See Data:K11a16/A-polynomial

[edit Notes for K11a16's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11a16's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+10 t^2-24 t+33-24 t^{-1} +10 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-2 z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 105, 0 }
Jones polynomial -q^5+3 q^4-6 q^3+11 q^2-14 q+17-17 q^{-1} +14 q^{-2} -11 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) -a^2 z^6-z^6+a^4 z^4-3 a^2 z^4+2 z^4 a^{-2} -2 z^4+2 a^4 z^2-5 a^2 z^2+4 z^2 a^{-2} -z^2 a^{-4} -2 z^2+2 a^4-3 a^2+3 a^{-2} - a^{-4}
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+4 a^3 z^9+7 a z^9+3 z^9 a^{-1} +5 a^4 z^8+9 a^2 z^8+4 z^8 a^{-2} +8 z^8+3 a^5 z^7-7 a^3 z^7-13 a z^7+z^7 a^{-1} +4 z^7 a^{-3} +a^6 z^6-14 a^4 z^6-31 a^2 z^6-z^6 a^{-2} +3 z^6 a^{-4} -20 z^6-8 a^5 z^5+2 a^3 z^5+11 a z^5-4 z^5 a^{-1} -4 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+13 a^4 z^4+37 a^2 z^4-7 z^4 a^{-2} -6 z^4 a^{-4} +20 z^4+5 a^5 z^3-2 a^3 z^3-8 a z^3-z^3 a^{-1} -2 z^3 a^{-3} -2 z^3 a^{-5} +2 a^6 z^2-8 a^4 z^2-20 a^2 z^2+7 z^2 a^{-2} +4 z^2 a^{-4} -7 z^2-a^5 z+2 a^3 z+5 a z+3 z a^{-1} +2 z a^{-3} +z a^{-5} +2 a^4+3 a^2-3 a^{-2} - a^{-4}
The A2 invariant Data:K11a16/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a16/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a280,}

Vassiliev invariants

V2 and V3: (-2, 3)
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 24 32 \frac{164}{3} \frac{100}{3} -192 -368 -96 -104 -\frac{256}{3} 288 -\frac{1312}{3} -\frac{800}{3} \frac{3089}{15} \frac{1148}{5} -\frac{16804}{45} \frac{1375}{9} -\frac{2191}{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=0 is the signature of K11a16. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
11           1-1
9          2 2
7         41 -3
5        72  5
3       74   -3
1      107    3
-1     88     0
-3    69      -3
-5   58       3
-7  26        -4
-9 15         4
-11 2          -2
-131           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\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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K11a15

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K11a17