K11a166

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K11a165.gif

K11a165

K11a167.gif

K11a167

Contents

K11a166.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a166 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X18,6,19,5 X16,8,17,7 X2,10,3,9 X22,11,1,12 X20,13,21,14 X8,16,9,15 X6,18,7,17 X14,19,15,20 X12,21,13,22
Gauss code 1, -5, 2, -1, 3, -9, 4, -8, 5, -2, 6, -11, 7, -10, 8, -4, 9, -3, 10, -7, 11, -6
Dowker-Thistlethwaite code 4 10 18 16 2 22 20 8 6 14 12
A Braid Representative
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A Morse Link Presentation K11a166 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a166/ThurstonBennequinNumber
Hyperbolic Volume 10.3671
A-Polynomial See Data:K11a166/A-polynomial

[edit Notes for K11a166's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a166's four dimensional invariants]

Polynomial invariants

Alexander polynomial -4 t^2+15 t-21+15 t^{-1} -4 t^{-2}
Conway polynomial -4 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 59, 2 }
Jones polynomial -q^8+2 q^7-3 q^6+6 q^5-8 q^4+9 q^3-9 q^2+8 q-6+4 q^{-1} -2 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -2 z^4 a^{-2} -z^4 a^{-4} -z^4+a^2 z^2-3 z^2 a^{-2} +2 z^2 a^{-6} -z^2+a^2- a^{-2} +2 a^{-6} - a^{-8}
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +4 z^9 a^{-3} +2 z^9 a^{-5} -2 z^8 a^{-2} -2 z^8 a^{-4} +2 z^8 a^{-6} +2 z^8+2 a z^7-7 z^7 a^{-1} -18 z^7 a^{-3} -7 z^7 a^{-5} +2 z^7 a^{-7} +a^2 z^6-4 z^6 a^{-6} +2 z^6 a^{-8} -5 z^6-7 a z^5+11 z^5 a^{-1} +35 z^5 a^{-3} +12 z^5 a^{-5} -4 z^5 a^{-7} +z^5 a^{-9} -4 a^2 z^4+6 z^4 a^{-2} +8 z^4 a^{-4} -6 z^4 a^{-8} +5 a z^3-12 z^3 a^{-1} -26 z^3 a^{-3} -6 z^3 a^{-5} -3 z^3 a^{-9} +4 a^2 z^2-8 z^2 a^{-2} -4 z^2 a^{-4} +6 z^2 a^{-6} +4 z^2 a^{-8} +2 z^2+4 z a^{-1} +6 z a^{-3} +2 z a^{-5} +z a^{-7} +z a^{-9} -a^2+ a^{-2} -2 a^{-6} - a^{-8}
The A2 invariant Data:K11a166/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a166/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_38,}

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

Vassiliev invariants

V2 and V3: (-1, 2)
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 16 8 \frac{322}{3} \frac{110}{3} -64 \frac{544}{3} -\frac{128}{3} 144 -\frac{32}{3} 128 -\frac{1288}{3} -\frac{440}{3} \frac{5969}{30} -\frac{5458}{15} \frac{13618}{45} \frac{3055}{18} \frac{209}{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=2 is the signature of K11a166. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          1 1
13         21 -1
11        41  3
9       42   -2
7      54    1
5     44     0
3    45      -1
1   35       2
-1  13        -2
-3 13         2
-5 1          -1
-71           1
Integral Khovanov Homology

(db, data source)

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