K11a202

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K11a201.gif

K11a201

K11a203.gif

K11a203

Contents

K11a202.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a202 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X16,6,17,5 X18,8,19,7 X14,10,15,9 X20,11,21,12 X2,13,3,14 X8,16,9,15 X6,18,7,17 X22,20,1,19 X10,21,11,22
Gauss code 1, -7, 2, -1, 3, -9, 4, -8, 5, -11, 6, -2, 7, -5, 8, -3, 9, -4, 10, -6, 11, -10
Dowker-Thistlethwaite code 4 12 16 18 14 20 2 8 6 22 10
A Braid Representative
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A Morse Link Presentation K11a202 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2,3\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a202/ThurstonBennequinNumber
Hyperbolic Volume 14.8711
A-Polynomial See Data:K11a202/A-polynomial

[edit Notes for K11a202'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 K11a202's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-12 t^2+26 t-31+26 t^{-1} -12 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6-4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 111, 2 }
Jones polynomial -q^8+4 q^7-8 q^6+13 q^5-16 q^4+18 q^3-18 q^2+14 q-10+6 q^{-1} -2 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +2 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-2 z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} -4 z^2+2 a^2-2 a^{-2} +2 a^{-4} -1
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +7 z^9 a^{-3} +4 z^9 a^{-5} +7 z^8 a^{-2} +11 z^8 a^{-4} +7 z^8 a^{-6} +3 z^8+2 a z^7-5 z^7 a^{-1} -11 z^7 a^{-3} +3 z^7 a^{-5} +7 z^7 a^{-7} +a^2 z^6-22 z^6 a^{-2} -27 z^6 a^{-4} -8 z^6 a^{-6} +4 z^6 a^{-8} -6 z^6-5 a z^5+6 z^5 a^{-1} +8 z^5 a^{-3} -15 z^5 a^{-5} -11 z^5 a^{-7} +z^5 a^{-9} -4 a^2 z^4+28 z^4 a^{-2} +27 z^4 a^{-4} -z^4 a^{-6} -6 z^4 a^{-8} +2 z^4+2 a z^3-10 z^3 a^{-1} -6 z^3 a^{-3} +11 z^3 a^{-5} +4 z^3 a^{-7} -z^3 a^{-9} +5 a^2 z^2-17 z^2 a^{-2} -13 z^2 a^{-4} +2 z^2 a^{-6} +2 z^2 a^{-8} +z^2+a z+6 z a^{-1} +4 z a^{-3} -2 z a^{-5} -z a^{-7} -2 a^2+2 a^{-2} +2 a^{-4} -1
The A2 invariant Data:K11a202/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a202/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a137,}

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

Vassiliev invariants

V2 and V3: (-4, -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
-16 -16 128 \frac{520}{3} \frac{200}{3} 256 \frac{1088}{3} \frac{224}{3} 48 -\frac{2048}{3} 128 -\frac{8320}{3} -\frac{3200}{3} -\frac{36062}{15} \frac{5048}{15} -\frac{92648}{45} \frac{2030}{9} -\frac{6542}{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 K11a202. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          3 3
13         51 -4
11        83  5
9       85   -3
7      108    2
5     88     0
3    610      -4
1   59       4
-1  15        -4
-3 15         4
-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}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11a201.gif

K11a201

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K11a203