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

Visit K11n19 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 X19,12,20,13 X21,14,22,15
Gauss code 1, -5, 2, -1, 3, 9, -4, -2, 5, -3, -6, 10, -7, 11, -8, 4, -9, 6, -10, 7, -11, 8
Dowker-Thistlethwaite code 4 8 10 -16 2 -18 -20 -22 -6 -12 -14
A Braid Representative
A Morse Link Presentation K11n19 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n19/ThurstonBennequinNumber
Hyperbolic Volume 4.76989
A-Polynomial See Data:K11n19/A-polynomial

[edit Notes for K11n19's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n19's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+2 t^2-1+2 t^{-2} - t^{-3}
Conway polynomial -z^6-4 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 5, -4 }
Jones polynomial q^2-q+1- q^{-1} + q^{-2}
HOMFLY-PT polynomial (db, data sources) -a^6+z^4 a^4+5 z^2 a^4+4 a^4-z^6 a^2-6 z^4 a^2-10 z^2 a^2-5 a^2+z^4+4 z^2+3
Kauffman polynomial (db, data sources) a^3 z^9+a z^9+a^4 z^8+2 a^2 z^8+z^8-7 a^3 z^7-7 a z^7-7 a^4 z^6-14 a^2 z^6-7 z^6+15 a^3 z^5+15 a z^5+15 a^4 z^4+30 a^2 z^4+15 z^4-a^5 z^3-12 a^3 z^3-11 a z^3-a^6 z^2-13 a^4 z^2-23 a^2 z^2-11 z^2+a^5 z+3 a^3 z+2 a z+a^6+4 a^4+5 a^2+3
The A2 invariant -q^{20}+q^{16}+q^{14}+q^{12}-q^6-q^4+ q^{-2} + q^{-4} + q^{-6}
The G2 invariant Data:K11n19/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n135,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {4_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{178}{3} \frac{110}{3} 0 -128 -32 -64 -\frac{32}{3} 0 -\frac{712}{3} -\frac{440}{3} -\frac{2431}{30} \frac{302}{15} -\frac{8702}{45} \frac{2335}{18} -\frac{1951}{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 K11n19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
5       11
3        0
1     11 0
-1   11   0
-3   11   0
-5 111    1
-7        0
-911      0
Integral Khovanov Homology

(db, data source)

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