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

Visit K11n117 at Knotilus!

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 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, 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 K11n117 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n117/ThurstonBennequinNumber
Hyperbolic Volume 11.3952
A-Polynomial See Data:K11n117/A-polynomial

[edit Notes for K11n117's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant -2

[edit Notes for K11n117's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^2+9 t-11+9 t^{-1} -3 t^{-2}
Conway polynomial -3 z^4-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 35, 2 }
Jones polynomial 2 q^5-4 q^4+5 q^3-6 q^2+6 q-5+4 q^{-1} -2 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -2 z^4 a^{-2} -z^4+a^2 z^2-5 z^2 a^{-2} +2 z^2 a^{-4} -z^2+a^2-3 a^{-2} +2 a^{-4} +1
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +3 z^8 a^{-2} +z^8 a^{-4} +2 z^8+2 a z^7-2 z^7 a^{-1} -4 z^7 a^{-3} +a^2 z^6-11 z^6 a^{-2} -4 z^6 a^{-4} -6 z^6-7 a z^5+z^5 a^{-1} +9 z^5 a^{-3} +z^5 a^{-5} -4 a^2 z^4+17 z^4 a^{-2} +10 z^4 a^{-4} +3 z^4+5 a z^3-4 z^3 a^{-1} -7 z^3 a^{-3} +2 z^3 a^{-5} +4 a^2 z^2-14 z^2 a^{-2} -7 z^2 a^{-4} +2 z^2 a^{-6} -z^2+2 z a^{-1} +z a^{-3} -z a^{-5} -a^2+3 a^{-2} +2 a^{-4} +1
The A2 invariant Data:K11n117/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n117/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_20, 10_162,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {10_138,}

Vassiliev invariants

V2 and V3: (-3, -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
-12 -16 72 114 62 192 \frac{1088}{3} \frac{224}{3} 112 -288 128 -1368 -744 -\frac{8831}{10} \frac{6106}{15} -\frac{19982}{15} \frac{1567}{6} -\frac{3551}{10}

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 K11n117. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11        22
9       2 -2
7      32 1
5     32  -1
3    33   0
1   34    1
-1  12     -1
-3 13      2
-5 1       -1
-71        1
Integral Khovanov Homology

(db, data source)

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

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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