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

Visit K11a222 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X18,5,19,6 X20,7,21,8 X14,10,15,9 X16,12,17,11 X2,13,3,14 X10,16,11,15 X22,17,1,18 X8,19,9,20 X6,21,7,22
Gauss code 1, -7, 2, -1, 3, -11, 4, -10, 5, -8, 6, -2, 7, -5, 8, -6, 9, -3, 10, -4, 11, -9
Dowker-Thistlethwaite code 4 12 18 20 14 16 2 10 22 8 6
A Braid Representative
A Morse Link Presentation K11a222 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a222/ThurstonBennequinNumber
Hyperbolic Volume 13.8903
A-Polynomial See Data:K11a222/A-polynomial

[edit Notes for K11a222'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 K11a222's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+11 t^2-23 t+29-23 t^{-1} +11 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 101, 0 }
Jones polynomial q^4-4 q^3+8 q^2-12 q+15-16 q^{-1} +16 q^{-2} -12 q^{-3} +9 q^{-4} -5 q^{-5} +2 q^{-6} - q^{-7}
HOMFLY-PT polynomial (db, data sources) -z^2 a^6-2 a^6+2 z^4 a^4+5 z^2 a^4+3 a^4-z^6 a^2-2 z^4 a^2-z^2 a^2-z^6-2 z^4-z^2+z^4 a^{-2} +z^2 a^{-2}
Kauffman polynomial (db, data sources) a^4 z^{10}+a^2 z^{10}+2 a^5 z^9+6 a^3 z^9+4 a z^9+2 a^6 z^8+3 a^4 z^8+9 a^2 z^8+8 z^8+a^7 z^7-4 a^5 z^7-12 a^3 z^7+3 a z^7+10 z^7 a^{-1} -8 a^6 z^6-16 a^4 z^6-24 a^2 z^6+8 z^6 a^{-2} -8 z^6-5 a^7 z^5-4 a^5 z^5+2 a^3 z^5-16 a z^5-13 z^5 a^{-1} +4 z^5 a^{-3} +10 a^6 z^4+17 a^4 z^4+15 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} -z^4+8 a^7 z^3+11 a^5 z^3+3 a^3 z^3+7 a z^3+5 z^3 a^{-1} -2 z^3 a^{-3} -5 a^6 z^2-8 a^4 z^2-4 a^2 z^2+2 z^2 a^{-2} +z^2-4 a^7 z-5 a^5 z-a^3 z+2 a^6+3 a^4
The A2 invariant Data:K11a222/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a222/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a204,}

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

Vassiliev invariants

V2 and V3: (3, -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
12 -40 72 190 42 -480 -\frac{2800}{3} -\frac{256}{3} -264 288 800 2280 504 \frac{46031}{10} -\frac{9586}{15} \frac{40342}{15} \frac{913}{6} \frac{4431}{10}

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 K11a222. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
9           11
7          3 -3
5         51 4
3        73  -4
1       85   3
-1      98    -1
-3     77     0
-5    59      4
-7   47       -3
-9  15        4
-11 14         -3
-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}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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