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

Visit K11n6 at Knotilus!

Knot K11n6.
A graph, knot K11n6.
A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X11,18,12,19 X6,14,7,13 X15,20,16,21 X17,12,18,13 X19,22,20,1 X21,16,22,17
Gauss code 1, -5, 2, -1, 3, -7, 4, -2, 5, -3, -6, 9, 7, -4, -8, 11, -9, 6, -10, 8, -11, 10
Dowker-Thistlethwaite code 4 8 10 14 2 -18 6 -20 -12 -22 -16
A Braid Representative
A Morse Link Presentation K11n6 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n6/ThurstonBennequinNumber
Hyperbolic Volume 11.407
A-Polynomial See Data:K11n6/A-polynomial

[edit Notes for K11n6's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11n6's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+3 t^2-3 t+3-3 t^{-1} +3 t^{-2} - t^{-3}
Conway polynomial -z^6-3 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 17, 0 }
Jones polynomial q^3-q^2+1-2 q^{-1} +3 q^{-2} -3 q^{-3} +4 q^{-4} -3 q^{-5} +2 q^{-6} - q^{-7}
HOMFLY-PT polynomial (db, data sources) -z^2 a^6-2 a^6+2 z^4 a^4+7 z^2 a^4+5 a^4-z^6 a^2-5 z^4 a^2-7 z^2 a^2-3 a^2+z^2 a^{-2} + a^{-2}
Kauffman polynomial (db, data sources) a^5 z^9+a^3 z^9+2 a^6 z^8+4 a^4 z^8+2 a^2 z^8+a^7 z^7-2 a^5 z^7-3 a^3 z^7+a z^7+z^7 a^{-1} -10 a^6 z^6-21 a^4 z^6-12 a^2 z^6+z^6 a^{-2} -5 a^7 z^5-8 a^5 z^5-5 a^3 z^5-8 a z^5-6 z^5 a^{-1} +14 a^6 z^4+31 a^4 z^4+19 a^2 z^4-5 z^4 a^{-2} -3 z^4+7 a^7 z^3+15 a^5 z^3+13 a^3 z^3+13 a z^3+8 z^3 a^{-1} -8 a^6 z^2-19 a^4 z^2-12 a^2 z^2+5 z^2 a^{-2} +4 z^2-3 a^7 z-6 a^5 z-6 a^3 z-6 a z-3 z a^{-1} +2 a^6+5 a^4+3 a^2- a^{-2}
The A2 invariant Data:K11n6/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n6/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_2,}

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

Vassiliev invariants

V2 and V3: (0, -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
0 -24 0 80 24 0 -112 0 -24 0 288 0 0 632 -\frac{40}{3} 200 56 8

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=0 is the signature of K11n6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7           11
5            0
3        111 -1
1       21   1
-1      221   -1
-3     221    1
-5    22      0
-7   221      1
-9  12        1
-11 12         -1
-13 1          1
-151           -1
Integral Khovanov Homology

(db, data source)

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