K11n90

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K11n89.gif

K11n89

K11n91.gif

K11n91

Contents

K11n90.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n90 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X16,8,17,7 X18,10,19,9 X2,11,3,12 X13,21,14,20 X8,16,9,15 X6,18,7,17 X19,1,20,22 X21,15,22,14
Gauss code 1, -6, 2, -1, 3, -9, 4, -8, 5, -2, 6, -3, -7, 11, 8, -4, 9, -5, -10, 7, -11, 10
Dowker-Thistlethwaite code 4 10 12 16 18 2 -20 8 6 -22 -14
A Braid Representative
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A Morse Link Presentation K11n90 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n90/ThurstonBennequinNumber
Hyperbolic Volume 11.0274
A-Polynomial See Data:K11n90/A-polynomial

[edit Notes for K11n90's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n90's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+7 t^2-8 t+7-8 t^{-1} +7 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-5 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \left\{2,t^2+t+1\right\}
Determinant and Signature { 41, 4 }
Jones polynomial -q^9+3 q^8-5 q^7+6 q^6-7 q^5+7 q^4-5 q^3+4 q^2-2 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -3 z^4 a^{-4} -4 z^4 a^{-6} +z^4 a^{-8} +3 z^2 a^{-2} -5 z^2 a^{-6} +4 z^2 a^{-8} + a^{-2} +2 a^{-4} -4 a^{-6} +3 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) z^9 a^{-5} +z^9 a^{-7} +2 z^8 a^{-4} +4 z^8 a^{-6} +2 z^8 a^{-8} +2 z^7 a^{-3} -z^7 a^{-5} -2 z^7 a^{-7} +z^7 a^{-9} +z^6 a^{-2} -6 z^6 a^{-4} -15 z^6 a^{-6} -8 z^6 a^{-8} -7 z^5 a^{-3} -3 z^5 a^{-5} +3 z^5 a^{-7} -z^5 a^{-9} -4 z^4 a^{-2} +4 z^4 a^{-4} +23 z^4 a^{-6} +18 z^4 a^{-8} +3 z^4 a^{-10} +5 z^3 a^{-3} +2 z^3 a^{-5} -z^3 a^{-7} +3 z^3 a^{-9} +z^3 a^{-11} +4 z^2 a^{-2} -3 z^2 a^{-4} -17 z^2 a^{-6} -13 z^2 a^{-8} -3 z^2 a^{-10} -z a^{-5} -z a^{-7} -z a^{-9} -z a^{-11} - a^{-2} +2 a^{-4} +4 a^{-6} +3 a^{-8} + a^{-10}
The A2 invariant Data:K11n90/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n90/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {9_20,}

Vassiliev invariants

V2 and V3: (2, 4)
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 32 32 \frac{556}{3} \frac{164}{3} 256 \frac{2912}{3} \frac{608}{3} 256 \frac{256}{3} 512 \frac{4448}{3} \frac{1312}{3} \frac{69511}{15} -\frac{8524}{15} \frac{137164}{45} \frac{1241}{9} \frac{6631}{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=4 is the signature of K11n90. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-101234567χ
19         1-1
17        2 2
15       31 -2
13      32  1
11     43   -1
9    33    0
7   24     2
5  23      -1
3 13       2
1 1        -1
-11         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11n89

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K11n91