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

Visit K11n20 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X7,17,8,16 X2,9,3,10 X11,18,12,19 X13,20,14,21 X15,22,16,1 X17,7,18,6 X19,14,20,15 X21,12,22,13
Gauss code 1, -5, 2, -1, 3, 9, -4, -2, 5, -3, -6, 11, -7, 10, -8, 4, -9, 6, -10, 7, -11, 8
Dowker-Thistlethwaite code 4 8 10 -16 2 -18 -20 -22 -6 -14 -12
A Braid Representative
A Morse Link Presentation K11n20 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n20/ThurstonBennequinNumber
Hyperbolic Volume 9.41057
A-Polynomial See Data:K11n20/A-polynomial

[edit Notes for K11n20'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 K11n20's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^2+6 t-7+6 t^{-1} -2 t^{-2}
Conway polynomial -2 z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 23, -2 }
Jones polynomial q^3-2 q^2+3 q-3+4 q^{-1} -4 q^{-2} +3 q^{-3} -2 q^{-4} + q^{-5}
HOMFLY-PT polynomial (db, data sources) z^2 a^4+a^4-z^4 a^2-2 z^2 a^2-a^2-z^4-2 z^2+z^2 a^{-2} + a^{-2}
Kauffman polynomial (db, data sources) a^3 z^9+a z^9+a^4 z^8+3 a^2 z^8+2 z^8-5 a^3 z^7-3 a z^7+2 z^7 a^{-1} -6 a^4 z^6-15 a^2 z^6+z^6 a^{-2} -8 z^6+7 a^3 z^5-a z^5-8 z^5 a^{-1} +12 a^4 z^4+22 a^2 z^4-4 z^4 a^{-2} +6 z^4+a^5 z^3-a^3 z^3+4 a z^3+6 z^3 a^{-1} -7 a^4 z^2-10 a^2 z^2+3 z^2 a^{-2} -a^5 z-a^3 z-a z-z a^{-1} +a^4+a^2- a^{-2}
The A2 invariant Data:K11n20/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n20/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_6, K11n151, K11n152,}

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

Vassiliev invariants

V2 and V3: (-2, 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
-8 8 32 \frac{164}{3} \frac{100}{3} -64 -\frac{400}{3} -\frac{64}{3} -56 -\frac{256}{3} 32 -\frac{1312}{3} -\frac{800}{3} -\frac{4111}{15} \frac{268}{5} -\frac{18244}{45} \frac{1231}{9} -\frac{1471}{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=-2 is the signature of K11n20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7        11
5       1 -1
3      21 1
1    121  0
-1    32   1
-3   22    0
-5  12     -1
-7 12      1
-9 1       -1
-111        1
Integral Khovanov Homology

(db, data source)

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