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

Visit K11n116 at Knotilus!

K11n116 is not k-colourable for any k. See The Determinant and the Signature.

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X5,14,6,15 X7,19,8,18 X9,17,10,16 X2,11,3,12 X20,13,21,14 X15,22,16,1 X17,9,18,8 X12,19,13,20 X21,7,22,6
Gauss code 1, -6, 2, -1, -3, 11, -4, 9, -5, -2, 6, -10, 7, 3, -8, 5, -9, 4, 10, -7, -11, 8
Dowker-Thistlethwaite code 4 10 -14 -18 -16 2 20 -22 -8 12 -6
A Braid Representative
A Morse Link Presentation K11n116 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n116/ThurstonBennequinNumber
Hyperbolic Volume 7.75445
A-Polynomial See Data:K11n116/A-polynomial

[edit Notes for K11n116's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n116's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^2+3- t^{-2}
Conway polynomial -z^4-4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 1, 0 }
Jones polynomial q^3-q^2+q- q^{-3} + q^{-4} - q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) a^6-z^2 a^4-a^4+a^2-z^4-4 z^2-2+z^2 a^{-2} +2 a^{-2}
Kauffman polynomial (db, data sources) a^5 z^7+z^7 a^{-1} +a^6 z^6+a^4 z^6+z^6 a^{-2} +z^6-5 a^5 z^5-5 z^5 a^{-1} -5 a^6 z^4-5 a^4 z^4-a^2 z^4-5 z^4 a^{-2} -6 z^4+6 a^5 z^3-a^3 z^3-2 a z^3+5 z^3 a^{-1} +6 a^6 z^2+5 a^4 z^2+a^2 z^2+6 z^2 a^{-2} +8 z^2-2 a^5 z+2 a z-a^6-a^4-a^2-2 a^{-2} -2
The A2 invariant Data:K11n116/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n116/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n49,}

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

Vassiliev invariants

V2 and V3: (-4, 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
-16 24 128 \frac{472}{3} \frac{224}{3} -384 -592 -128 -136 -\frac{2048}{3} 288 -\frac{7552}{3} -\frac{3584}{3} -\frac{22022}{15} \frac{8128}{15} -\frac{87728}{45} \frac{3158}{9} -\frac{7622}{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=0 is the signature of K11n116. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7          11
5           0
3        11 0
1      21   1
-1      11   0
-3    121    0
-5   1       -1
-7   11      0
-9 11        0
-11           0
-131          1
Integral Khovanov Homology

(db, data source)

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