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

Visit K11n111 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X5,14,6,15 X7,17,8,16 X9,19,10,18 X2,11,3,12 X13,21,14,20 X15,22,16,1 X17,9,18,8 X19,13,20,12 X21,7,22,6
Gauss code 1, -6, 2, -1, -3, 11, -4, 9, -5, -2, 6, 10, -7, 3, -8, 4, -9, 5, -10, 7, -11, 8
Dowker-Thistlethwaite code 4 10 -14 -16 -18 2 -20 -22 -8 -12 -6
A Braid Representative
A Morse Link Presentation K11n111 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:K11n111/ThurstonBennequinNumber
Hyperbolic Volume 8.61249
A-Polynomial See Data:K11n111/A-polynomial

[edit Notes for K11n111'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 K11n111's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+t^2+3 t-5+3 t^{-1} + t^{-2} - t^{-3}
Conway polynomial -z^6-5 z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 7, 2 }
Jones polynomial q^7-2 q^6+2 q^5-2 q^4+2 q^3-q^2+1- q^{-1} + q^{-2}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -6 z^4 a^{-2} +z^4-9 z^2 a^{-2} +2 z^2 a^{-4} +z^2 a^{-6} +4 z^2-4 a^{-2} +2 a^{-4} +3
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +2 z^8 a^{-2} +z^8 a^{-4} +z^8-7 z^7 a^{-1} -7 z^7 a^{-3} -15 z^6 a^{-2} -7 z^6 a^{-4} +z^6 a^{-6} -7 z^6+14 z^5 a^{-1} +13 z^5 a^{-3} +z^5 a^{-5} +2 z^5 a^{-7} +32 z^4 a^{-2} +14 z^4 a^{-4} -2 z^4 a^{-6} +z^4 a^{-8} +15 z^4-9 z^3 a^{-1} -6 z^3 a^{-3} -2 z^3 a^{-5} -5 z^3 a^{-7} -22 z^2 a^{-2} -8 z^2 a^{-4} -2 z^2 a^{-8} -12 z^2+z a^{-1} +z a^{-3} +z a^{-5} +z a^{-7} +4 a^{-2} +2 a^{-4} +3
The A2 invariant q^6+q^4+q^2+ q^{-2} - q^{-4} - q^{-6} - q^{-10} + q^{-12} + q^{-16} - q^{-20} + q^{-22}
The G2 invariant Data:K11n111/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: (-2, -2)
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 -16 32 \frac{212}{3} \frac{172}{3} 128 \frac{992}{3} \frac{320}{3} 112 -\frac{256}{3} 128 -\frac{1696}{3} -\frac{1376}{3} \frac{89}{15} \frac{6044}{15} -\frac{32284}{45} \frac{2023}{9} -\frac{3751}{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 K11n111. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
15          11
13         1 -1
11        11 0
9      121  0
7      11   0
5    122    1
3   111     -1
1   12      1
-1 11        0
-3           0
-51          1
Integral Khovanov Homology

(db, data source)

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