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

Visit K11n82 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X5,17,6,16 X7,12,8,13 X2,9,3,10 X11,18,12,19 X13,21,14,20 X15,1,16,22 X17,10,18,11 X19,7,20,6 X21,15,22,14
Gauss code 1, -5, 2, -1, -3, 10, -4, -2, 5, 9, -6, 4, -7, 11, -8, 3, -9, 6, -10, 7, -11, 8
Dowker-Thistlethwaite code 4 8 -16 -12 2 -18 -20 -22 -10 -6 -14
A Braid Representative
A Morse Link Presentation K11n82 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n82/ThurstonBennequinNumber
Hyperbolic Volume 8.68999
A-Polynomial See Data:K11n82/A-polynomial

[edit Notes for K11n82'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 K11n82's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-3 t^2+4 t-3+4 t^{-1} -3 t^{-2} + t^{-3}
Conway polynomial z^6+3 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 19, 2 }
Jones polynomial -q^4+2 q^3-2 q^2+3 q-3+3 q^{-1} -2 q^{-2} +2 q^{-3} - q^{-4}
HOMFLY-PT polynomial (db, data sources) z^6-a^2 z^4-z^4 a^{-2} +5 z^4-3 a^2 z^2-3 z^2 a^{-2} +7 z^2-a^2- a^{-2} +3
Kauffman polynomial (db, data sources) a z^9+z^9 a^{-1} +2 a^2 z^8+z^8 a^{-2} +3 z^8+a^3 z^7-4 a z^7-5 z^7 a^{-1} -11 a^2 z^6-5 z^6 a^{-2} -16 z^6-5 a^3 z^5+2 a z^5+8 z^5 a^{-1} +z^5 a^{-3} +17 a^2 z^4+8 z^4 a^{-2} +25 z^4+6 a^3 z^3+2 a z^3-7 z^3 a^{-1} -3 z^3 a^{-3} -8 a^2 z^2-6 z^2 a^{-2} +z^2 a^{-4} -15 z^2-a^3 z-a z+z a^{-1} +2 z a^{-3} +z a^{-5} +a^2+ a^{-2} +3
The A2 invariant -q^{12}+q^6+q^4+1+ q^{-4} + q^{-6} - q^{-12}
The G2 invariant Data:K11n82/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (1, 0)
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 0 8 -\frac{34}{3} -\frac{62}{3} 0 -32 32 -64 \frac{32}{3} 0 -\frac{136}{3} -\frac{248}{3} -\frac{2609}{30} \frac{806}{5} -\frac{10978}{45} -\frac{559}{18} -\frac{2129}{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 K11n82. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
9        1-1
7       1 1
5      11 0
3     21  1
1    22   0
-1   11    0
-3  12     1
-5 11      0
-7 1       1
-91        -1
Integral Khovanov Homology

(db, data source)

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