K11a118

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K11a117.gif

K11a117

K11a119.gif

K11a119

Contents

K11a118.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a118 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X14,6,15,5 X18,7,19,8 X12,10,13,9 X2,11,3,12 X8,14,9,13 X20,16,21,15 X22,18,1,17 X6,19,7,20 X16,22,17,21
Gauss code 1, -6, 2, -1, 3, -10, 4, -7, 5, -2, 6, -5, 7, -3, 8, -11, 9, -4, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 10 14 18 12 2 8 20 22 6 16
A Braid Representative
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A Morse Link Presentation K11a118 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:K11a118/ThurstonBennequinNumber
Hyperbolic Volume 13.3497
A-Polynomial See Data:K11a118/A-polynomial

[edit Notes for K11a118's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for K11a118's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-10 t^2+20 t-23+20 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 { 87, 2 }
Jones polynomial -q^8+3 q^7-6 q^6+10 q^5-12 q^4+14 q^3-14 q^2+11 q-8+5 q^{-1} -2 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2+4 z^2 a^{-4} -2 z^2 a^{-6} -5 z^2+2 a^2- a^{-2} +3 a^{-4} - a^{-6} -2
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +6 z^9 a^{-3} +4 z^9 a^{-5} +2 z^8 a^{-2} +6 z^8 a^{-4} +6 z^8 a^{-6} +2 z^8+2 a z^7-3 z^7 a^{-1} -19 z^7 a^{-3} -9 z^7 a^{-5} +5 z^7 a^{-7} +a^2 z^6-9 z^6 a^{-2} -26 z^6 a^{-4} -17 z^6 a^{-6} +3 z^6 a^{-8} -2 z^6-6 a z^5+3 z^5 a^{-1} +31 z^5 a^{-3} +10 z^5 a^{-5} -11 z^5 a^{-7} +z^5 a^{-9} -4 a^2 z^4+13 z^4 a^{-2} +43 z^4 a^{-4} +21 z^4 a^{-6} -6 z^4 a^{-8} -7 z^4+4 a z^3-9 z^3 a^{-1} -23 z^3 a^{-3} -2 z^3 a^{-5} +6 z^3 a^{-7} -2 z^3 a^{-9} +5 a^2 z^2-10 z^2 a^{-2} -24 z^2 a^{-4} -10 z^2 a^{-6} +z^2 a^{-8} +8 z^2+5 z a^{-1} +7 z a^{-3} +z a^{-5} -z a^{-7} -2 a^2+ a^{-2} +3 a^{-4} + a^{-6} -2
The A2 invariant Data:K11a118/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a118/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a90,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a45,}

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{4}{3} -64 -\frac{400}{3} -\frac{352}{3} 8 -\frac{256}{3} 32 -\frac{1312}{3} -\frac{32}{3} -\frac{8431}{15} -\frac{7276}{15} \frac{9596}{45} -\frac{257}{9} \frac{1169}{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 K11a118. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          2 2
13         41 -3
11        62  4
9       64   -2
7      86    2
5     66     0
3    58      -3
1   47       3
-1  14        -3
-3 14         3
-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}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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K11a117.gif

K11a117

K11a119.gif

K11a119