K11a90

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K11a89.gif

K11a89

K11a91.gif

K11a91

Contents

K11a90.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a90 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X16,7,17,8 X18,9,19,10 X2,11,3,12 X22,14,1,13 X20,16,21,15 X8,17,9,18 X6,19,7,20 X14,22,15,21
Gauss code 1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, -8, 11, -7
Dowker-Thistlethwaite code 4 10 12 16 18 2 22 20 8 6 14
A Braid Representative
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A Morse Link Presentation K11a90 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a90/ThurstonBennequinNumber
Hyperbolic Volume 12.5188
A-Polynomial See Data:K11a90/A-polynomial

[edit Notes for K11a90'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 K11a90'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^4+3 q^3-5 q^2+9 q-11+13 q^{-1} -14 q^{-2} +12 q^{-3} -9 q^{-4} +6 q^{-5} -3 q^{-6} + q^{-7}
HOMFLY-PT polynomial (db, data sources) z^2 a^6+a^6-2 z^4 a^4-4 z^2 a^4-a^4+z^6 a^2+2 z^4 a^2-a^2+z^6+3 z^4+3 z^2+2-z^4 a^{-2} -2 z^2 a^{-2}
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+3 a^3 z^9+6 a z^9+3 z^9 a^{-1} +5 a^4 z^8+6 a^2 z^8+3 z^8 a^{-2} +4 z^8+6 a^5 z^7+a^3 z^7-16 a z^7-10 z^7 a^{-1} +z^7 a^{-3} +5 a^6 z^6-5 a^4 z^6-21 a^2 z^6-13 z^6 a^{-2} -24 z^6+3 a^7 z^5-9 a^5 z^5-14 a^3 z^5+10 a z^5+8 z^5 a^{-1} -4 z^5 a^{-3} +a^8 z^4-6 a^6 z^4-2 a^4 z^4+18 a^2 z^4+17 z^4 a^{-2} +30 z^4-3 a^7 z^3+8 a^5 z^3+14 a^3 z^3-3 a z^3-2 z^3 a^{-1} +4 z^3 a^{-3} -a^8 z^2+3 a^6 z^2+5 a^4 z^2-7 a^2 z^2-7 z^2 a^{-2} -15 z^2-3 a^5 z-3 a^3 z+a z+z a^{-1} -a^6-a^4+a^2+2
The A2 invariant Data:K11a90/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a90/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a118,}

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

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{28}{3} \frac{4}{3} -192 -240 -128 24 -\frac{256}{3} 288 \frac{224}{3} -\frac{32}{3} \frac{11729}{15} \frac{268}{5} \frac{16796}{45} -\frac{401}{9} \frac{929}{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 K11a90. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
9           1-1
7          2 2
5         31 -2
3        62  4
1       53   -2
-1      86    2
-3     76     -1
-5    57      -2
-7   47       3
-9  25        -3
-11 14         3
-13 2          -2
-151           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11a89

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