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

Visit K11n69 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8493 X5,15,6,14 X2837 X9,19,10,18 X11,17,12,16 X13,20,14,21 X15,7,16,6 X17,11,18,10 X19,1,20,22 X21,12,22,13
Gauss code 1, -4, 2, -1, -3, 8, 4, -2, -5, 9, -6, 11, -7, 3, -8, 6, -9, 5, -10, 7, -11, 10
Dowker-Thistlethwaite code 4 8 -14 2 -18 -16 -20 -6 -10 -22 -12
A Braid Representative
A Morse Link Presentation K11n69 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n69/ThurstonBennequinNumber
Hyperbolic Volume 11.956
A-Polynomial See Data:K11n69/A-polynomial

[edit Notes for K11n69's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [2,3]
Rasmussen s-Invariant -4

[edit Notes for K11n69's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+7 t^2-9 t+9-9 t^{-1} +7 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-5 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 45, 4 }
Jones polynomial -q^9+3 q^8-5 q^7+7 q^6-8 q^5+7 q^4-6 q^3+5 q^2-2 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -3 z^4 a^{-4} -4 z^4 a^{-6} +z^4 a^{-8} +3 z^2 a^{-2} -z^2 a^{-4} -5 z^2 a^{-6} +4 z^2 a^{-8} +2 a^{-2} -3 a^{-6} +3 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) z^9 a^{-5} +z^9 a^{-7} +2 z^8 a^{-4} +4 z^8 a^{-6} +2 z^8 a^{-8} +2 z^7 a^{-3} -z^7 a^{-7} +z^7 a^{-9} +z^6 a^{-2} -4 z^6 a^{-4} -12 z^6 a^{-6} -7 z^6 a^{-8} -6 z^5 a^{-3} -4 z^5 a^{-5} +z^5 a^{-7} -z^5 a^{-9} -4 z^4 a^{-2} -2 z^4 a^{-4} +15 z^4 a^{-6} +16 z^4 a^{-8} +3 z^4 a^{-10} +3 z^3 a^{-3} -z^3 a^{-5} -z^3 a^{-7} +4 z^3 a^{-9} +z^3 a^{-11} +5 z^2 a^{-2} +2 z^2 a^{-4} -12 z^2 a^{-6} -12 z^2 a^{-8} -3 z^2 a^{-10} +z a^{-3} +2 z a^{-5} +z a^{-7} -z a^{-9} -z a^{-11} -2 a^{-2} +3 a^{-6} +3 a^{-8} + a^{-10}
The A2 invariant Data:K11n69/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n69/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_21,}

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

Vassiliev invariants

V2 and V3: (1, 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
4 16 8 \frac{350}{3} \frac{130}{3} 64 \frac{1792}{3} \frac{448}{3} 176 \frac{32}{3} 128 \frac{1400}{3} \frac{520}{3} \frac{72271}{30} -\frac{954}{5} \frac{71582}{45} \frac{1937}{18} \frac{5551}{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=4 is the signature of K11n69. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
19         1-1
17        2 2
15       31 -2
13      42  2
11     43   -1
9    34    -1
7   34     1
5  23      -1
3 14       3
1 1        -1
-11         1
Integral Khovanov Homology

(db, data source)

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