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

Visit K11n65 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8493 X14,5,15,6 X2837 X18,9,19,10 X16,11,17,12 X13,20,14,21 X6,15,7,16 X10,17,11,18 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 K11n65 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2,3\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n65/ThurstonBennequinNumber
Hyperbolic Volume 11.4167
A-Polynomial See Data:K11n65/A-polynomial

[edit Notes for K11n65's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant 0

[edit Notes for K11n65's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^2-8 t+11-8 t^{-1} +3 t^{-2}
Conway polynomial 3 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 33, 0 }
Jones polynomial -2 q+4-4 q^{-1} +6 q^{-2} -5 q^{-3} +5 q^{-4} -4 q^{-5} +2 q^{-6} - q^{-7}
HOMFLY-PT polynomial (db, data sources) -z^2 a^6-a^6+z^4 a^4+z^2 a^4-a^4+2 z^4 a^2+6 z^2 a^2+5 a^2-2 z^2-2
Kauffman polynomial (db, data sources) a^5 z^9+a^3 z^9+2 a^6 z^8+4 a^4 z^8+2 a^2 z^8+a^7 z^7-a^5 z^7-a^3 z^7+a z^7-9 a^6 z^6-15 a^4 z^6-6 a^2 z^6-5 a^7 z^5-9 a^5 z^5-5 a^3 z^5-a z^5+11 a^6 z^4+12 a^4 z^4+2 a^2 z^4+z^4+8 a^7 z^3+13 a^5 z^3+2 a^3 z^3-3 a z^3-4 a^6 z^2-a^4 z^2+5 a^2 z^2+2 z^2-4 a^7 z-4 a^5 z+2 a^3 z+4 a z+2 z a^{-1} +a^6-a^4-5 a^2-2
The A2 invariant Data:K11n65/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n65/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_15,}

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

Vassiliev invariants

V2 and V3: (4, -5)
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
16 -40 128 \frac{680}{3} \frac{112}{3} -640 -\frac{3184}{3} -\frac{448}{3} -200 \frac{2048}{3} 800 \frac{10880}{3} \frac{1792}{3} \frac{85862}{15} -\frac{1808}{15} \frac{114848}{45} \frac{682}{9} \frac{5222}{15}

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 K11n65. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
3        2-2
1       2 2
-1      33 0
-3     31  2
-5    23   1
-7   33    0
-9  12     1
-11 13      -2
-13 1       1
-151        -1
Integral Khovanov Homology

(db, data source)

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