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

Visit K11n106 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X5,15,6,14 X7,16,8,17 X2,10,3,9 X20,11,21,12 X22,13,1,14 X15,18,16,19 X17,8,18,9 X19,7,20,6 X12,21,13,22
Gauss code 1, -5, 2, -1, -3, 10, -4, 9, 5, -2, 6, -11, 7, 3, -8, 4, -9, 8, -10, -6, 11, -7
Dowker-Thistlethwaite code 4 10 -14 -16 2 20 22 -18 -8 -6 12
A Braid Representative
A Morse Link Presentation K11n106 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n106/ThurstonBennequinNumber
Hyperbolic Volume 9.99629
A-Polynomial See Data:K11n106/A-polynomial

[edit Notes for K11n106's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 1
Rasmussen s-Invariant 2

[edit Notes for K11n106's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-3 t^2+6 t-7+6 t^{-1} -3 t^{-2} + t^{-3}
Conway polynomial z^6+3 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 27, -2 }
Jones polynomial -q^4+2 q^3-3 q^2+4 q-4+5 q^{-1} -3 q^{-2} +3 q^{-3} -2 q^{-4}
HOMFLY-PT polynomial (db, data sources) z^6-a^2 z^4-z^4 a^{-2} +5 z^4-2 a^2 z^2-3 z^2 a^{-2} +8 z^2-a^4-2 a^{-2} +4
Kauffman polynomial (db, data sources) a z^9+z^9 a^{-1} +2 a^2 z^8+2 z^8 a^{-2} +4 z^8+a^3 z^7-2 a z^7-2 z^7 a^{-1} +z^7 a^{-3} -9 a^2 z^6-10 z^6 a^{-2} -19 z^6-3 a^3 z^5-5 a z^5-7 z^5 a^{-1} -5 z^5 a^{-3} +a^4 z^4+13 a^2 z^4+14 z^4 a^{-2} +26 z^4+2 a^3 z^3+9 a z^3+14 z^3 a^{-1} +7 z^3 a^{-3} -8 a^2 z^2-7 z^2 a^{-2} -15 z^2+2 a^5 z+a^3 z-4 a z-6 z a^{-1} -3 z a^{-3} -a^4+2 a^{-2} +4
The A2 invariant -q^{14}-2 q^{12}+2 q^6+2 q^4+q^2+2+ q^{-4} - q^{-8} - q^{-12}
The G2 invariant Data:K11n106/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_10, 10_143,}

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

Vassiliev invariants

V2 and V3: (3, -1)
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 -8 72 94 10 -96 -\frac{464}{3} -\frac{128}{3} -8 288 32 1128 120 \frac{12591}{10} \frac{1534}{15} \frac{5942}{15} \frac{17}{6} \frac{271}{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 K11n106. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
9        1-1
7       1 1
5      21 -1
3     21  1
1    22   0
-1   32    1
-3  13     2
-5 22      0
-7 1       1
-92        -2
Integral Khovanov Homology

(db, data source)

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