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

Visit K11n84 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X5,18,6,19 X7,12,8,13 X2,9,3,10 X11,17,12,16 X13,20,14,21 X15,6,16,7 X17,11,18,10 X19,22,20,1 X21,14,22,15
Gauss code 1, -5, 2, -1, -3, 8, -4, -2, 5, 9, -6, 4, -7, 11, -8, 6, -9, 3, -10, 7, -11, 10
Dowker-Thistlethwaite code 4 8 -18 -12 2 -16 -20 -6 -10 -22 -14
A Braid Representative
A Morse Link Presentation K11n84 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:K11n84/ThurstonBennequinNumber
Hyperbolic Volume 11.4316
A-Polynomial See Data:K11n84/A-polynomial

[edit Notes for K11n84's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n84's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^2+9 t-13+9 t^{-1} -2 t^{-2}
Conway polynomial -2 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 35, -2 }
Jones polynomial 2 q^{-1} -3 q^{-2} +5 q^{-3} -6 q^{-4} +6 q^{-5} -5 q^{-6} +4 q^{-7} -3 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8-z^4 a^6-z^2 a^6-z^4 a^4-z^2 a^4-a^4+2 z^2 a^2+2 a^2
Kauffman polynomial (db, data sources) z^6 a^{10}-3 z^4 a^{10}+z^2 a^{10}+3 z^7 a^9-11 z^5 a^9+8 z^3 a^9-z a^9+3 z^8 a^8-11 z^6 a^8+9 z^4 a^8-2 z^2 a^8+z^9 a^7-10 z^5 a^7+11 z^3 a^7-3 z a^7+4 z^8 a^6-16 z^6 a^6+19 z^4 a^6-6 z^2 a^6+z^9 a^5-3 z^7 a^5+2 z^5 a^5+3 z^3 a^5-2 z a^5+z^8 a^4-4 z^6 a^4+7 z^4 a^4-z^2 a^4-a^4+z^5 a^3+2 z^2 a^2-2 a^2
The A2 invariant Data:K11n84/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n84/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_12,}

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

Vassiliev invariants

V2 and V3: (1, -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
4 -8 8 \frac{110}{3} \frac{58}{3} -32 -\frac{464}{3} -\frac{224}{3} -40 \frac{32}{3} 32 \frac{440}{3} \frac{232}{3} \frac{15871}{30} \frac{458}{15} \frac{17942}{45} \frac{65}{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=-2 is the signature of K11n84. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
-1        22
-3       21-1
-5      31 2
-7     32  -1
-9    33   0
-11   23    1
-13  23     -1
-15 12      1
-17 2       -2
-191        1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}\oplus{\mathbb Z}_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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