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

Visit K11n165 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X3,11,4,10 X14,5,15,6 X16,8,17,7 X20,9,21,10 X11,5,12,4 X18,13,19,14 X2,15,3,16 X22,18,1,17 X12,19,13,20 X8,21,9,22
Gauss code 1, -8, -2, 6, 3, -1, 4, -11, 5, 2, -6, -10, 7, -3, 8, -4, 9, -7, 10, -5, 11, -9
Dowker-Thistlethwaite code 6 -10 14 16 20 -4 18 2 22 12 8
A Braid Representative
A Morse Link Presentation K11n165 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n165/ThurstonBennequinNumber
Hyperbolic Volume 15.8169
A-Polynomial See Data:K11n165/A-polynomial

[edit Notes for K11n165'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 K11n165's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+7 t^2-20 t+29-20 t^{-1} +7 t^{-2} - t^{-3}
Conway polynomial -z^6+z^4-z^2+1
2nd Alexander ideal (db, data sources) \left\{2,t^2+t+1\right\}
Determinant and Signature { 85, 0 }
Jones polynomial -q^5+4 q^4-8 q^3+12 q^2-14 q+15-13 q^{-1} +10 q^{-2} -6 q^{-3} +2 q^{-4}
HOMFLY-PT polynomial (db, data sources) -z^6+a^2 z^4+2 z^4 a^{-2} -2 z^4-a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} -z^2+a^4-2 a^2+2
Kauffman polynomial (db, data sources) 2 a z^9+2 z^9 a^{-1} +3 a^2 z^8+6 z^8 a^{-2} +9 z^8+a^3 z^7+2 a z^7+8 z^7 a^{-1} +7 z^7 a^{-3} -2 a^2 z^6-6 z^6 a^{-2} +4 z^6 a^{-4} -12 z^6+5 a^3 z^5+a z^5-17 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} +3 a^4 z^4+5 a^2 z^4-3 z^4 a^{-2} -6 z^4 a^{-4} +5 z^4-7 a^3 z^3-8 a z^3+5 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} -3 a^4 z^2-7 a^2 z^2+2 z^2 a^{-2} +2 z^2 a^{-4} -4 z^2+2 a^3 z+3 a z+z a^{-1} +a^4+2 a^2+2
The A2 invariant Data:K11n165/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n165/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_60,}

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

Vassiliev invariants

V2 and V3: (-1, 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
-4 8 8 -\frac{14}{3} -\frac{10}{3} -32 -\frac{112}{3} -\frac{160}{3} 40 -\frac{32}{3} 32 \frac{56}{3} \frac{40}{3} \frac{2849}{30} \frac{742}{15} \frac{538}{45} -\frac{161}{18} -\frac{511}{30}

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 K11n165. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11         1-1
9        3 3
7       51 -4
5      73  4
3     75   -2
1    87    1
-1   68     2
-3  47      -3
-5 26       4
-7 4        -4
-92         2
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-4 {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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