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

Visit K11a69 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X16,5,17,6 X14,8,15,7 X2,9,3,10 X18,12,19,11 X20,14,21,13 X22,15,1,16 X12,18,13,17 X10,20,11,19 X6,21,7,22
Gauss code 1, -5, 2, -1, 3, -11, 4, -2, 5, -10, 6, -9, 7, -4, 8, -3, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 8 16 14 2 18 20 22 12 10 6
A Braid Representative
A Morse Link Presentation K11a69 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a69/ThurstonBennequinNumber
Hyperbolic Volume 16.7403
A-Polynomial See Data:K11a69/A-polynomial

[edit Notes for K11a69'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 K11a69's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+13 t^2-33 t+45-33 t^{-1} +13 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6+z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 141, 0 }
Jones polynomial q^6-4 q^5+8 q^4-14 q^3+20 q^2-22 q+23-20 q^{-1} +15 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -z^6+2 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -a^4 z^2+a^2 z^2-3 z^2 a^{-2} +z^2 a^{-4} +3 z^2-a^2- a^{-2} +3
Kauffman polynomial (db, data sources) 2 z^{10} a^{-2} +2 z^{10}+6 a z^9+12 z^9 a^{-1} +6 z^9 a^{-3} +9 a^2 z^8+12 z^8 a^{-2} +7 z^8 a^{-4} +14 z^8+8 a^3 z^7+3 a z^7-17 z^7 a^{-1} -8 z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-10 a^2 z^6-37 z^6 a^{-2} -18 z^6 a^{-4} +z^6 a^{-6} -32 z^6+a^5 z^5-12 a^3 z^5-18 a z^5+z^5 a^{-1} -4 z^5 a^{-3} -10 z^5 a^{-5} -5 a^4 z^4+2 a^2 z^4+34 z^4 a^{-2} +15 z^4 a^{-4} -2 z^4 a^{-6} +24 z^4-a^5 z^3+7 a^3 z^3+11 a z^3+5 z^3 a^{-1} +8 z^3 a^{-3} +6 z^3 a^{-5} +2 a^4 z^2-a^2 z^2-13 z^2 a^{-2} -4 z^2 a^{-4} -12 z^2-a^3 z-2 a z-2 z a^{-1} -z a^{-3} +a^2+ a^{-2} +3
The A2 invariant Data:K11a69/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a69/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: (1, 0)
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 0 8 \frac{14}{3} -\frac{14}{3} 0 0 64 -64 \frac{32}{3} 0 \frac{56}{3} -\frac{56}{3} \frac{511}{30} \frac{1858}{15} -\frac{7618}{45} \frac{641}{18} -\frac{1409}{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 K11a69. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
13           11
11          3 -3
9         51 4
7        93  -6
5       115   6
3      119    -2
1     1211     1
-1    912      3
-3   611       -5
-5  39        6
-7 16         -5
-9 3          3
-111           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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