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

Visit K11n122 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X5,16,6,17 X7,12,8,13 X9,19,10,18 X2,11,3,12 X13,20,14,21 X15,6,16,7 X17,22,18,1 X19,9,20,8 X21,14,22,15
Gauss code 1, -6, 2, -1, -3, 8, -4, 10, -5, -2, 6, 4, -7, 11, -8, 3, -9, 5, -10, 7, -11, 9
Dowker-Thistlethwaite code 4 10 -16 -12 -18 2 -20 -6 -22 -8 -14
A Braid Representative
A Morse Link Presentation K11n122 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n122/ThurstonBennequinNumber
Hyperbolic Volume 9.7305
A-Polynomial See Data:K11n122/A-polynomial

[edit Notes for K11n122's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 1
Rasmussen s-Invariant 2

[edit Notes for K11n122's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^2+7 t-9+7 t^{-1} -2 t^{-2}
Conway polynomial -2 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 27, -2 }
Jones polynomial 2 q^{-1} -2 q^{-2} +4 q^{-3} -5 q^{-4} +4 q^{-5} -4 q^{-6} +3 q^{-7} -2 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8+a^8-z^4 a^6-2 z^2 a^6-a^6-z^4 a^4-2 z^2 a^4-2 a^4+2 z^2 a^2+3 a^2
Kauffman polynomial (db, data sources) z^6 a^{10}-4 z^4 a^{10}+3 z^2 a^{10}+2 z^7 a^9-8 z^5 a^9+7 z^3 a^9-2 z a^9+2 z^8 a^8-8 z^6 a^8+8 z^4 a^8-4 z^2 a^8+a^8+z^9 a^7-3 z^7 a^7+z^5 a^7+3 z^8 a^6-14 z^6 a^6+21 z^4 a^6-11 z^2 a^6+a^6+z^9 a^5-5 z^7 a^5+10 z^5 a^5-9 z^3 a^5+5 z a^5+z^8 a^4-5 z^6 a^4+9 z^4 a^4-2 z^2 a^4-2 a^4+z^5 a^3-2 z^3 a^3+3 z a^3+2 z^2 a^2-3 a^2
The A2 invariant Data:K11n122/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n122/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_11, 10_147,}

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

Vassiliev invariants

V2 and V3: (-1, 4)
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 32 8 -\frac{206}{3} \frac{62}{3} -128 -\frac{448}{3} -\frac{256}{3} -128 -\frac{32}{3} 512 \frac{824}{3} -\frac{248}{3} \frac{50369}{30} -\frac{2818}{15} \frac{49138}{45} \frac{3295}{18} \frac{2849}{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 K11n122. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
-1        22
-3       110
-5      31 2
-7     21  -1
-9    23   -1
-11   22    0
-13  12     -1
-15 12      1
-17 1       -1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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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