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

Visit K11n114 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X14,6,15,5 X18,7,19,8 X16,9,17,10 X2,11,3,12 X13,21,14,20 X22,16,1,15 X8,17,9,18 X19,13,20,12 X6,21,7,22
Gauss code 1, -6, 2, -1, 3, -11, 4, -9, 5, -2, 6, 10, -7, -3, 8, -5, 9, -4, -10, 7, 11, -8
Dowker-Thistlethwaite code 4 10 14 18 16 2 -20 22 8 -12 6
A Braid Representative
A Morse Link Presentation K11n114 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n114/ThurstonBennequinNumber
Hyperbolic Volume 12.8914
A-Polynomial See Data:K11n114/A-polynomial

[edit Notes for K11n114'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 K11n114's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^2-13 t+21-13 t^{-1} +3 t^{-2}
Conway polynomial 3 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 53, 0 }
Jones polynomial -q^5+3 q^4-5 q^3+8 q^2-9 q+9-8 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) a^4-3 z^2 a^2-2 a^2+2 z^4+3 z^2+2+z^4 a^{-2} -z^2 a^{-4}
Kauffman polynomial (db, data sources) a z^9+z^9 a^{-1} +a^2 z^8+3 z^8 a^{-2} +4 z^8-2 a z^7+2 z^7 a^{-1} +4 z^7 a^{-3} -2 a^2 z^6-5 z^6 a^{-2} +3 z^6 a^{-4} -10 z^6+3 a^3 z^5+7 a z^5-6 z^5 a^{-1} -9 z^5 a^{-3} +z^5 a^{-5} +a^4 z^4+8 a^2 z^4+2 z^4 a^{-2} -7 z^4 a^{-4} +16 z^4-4 a^3 z^3-5 a z^3+6 z^3 a^{-1} +5 z^3 a^{-3} -2 z^3 a^{-5} -2 a^4 z^2-9 a^2 z^2+3 z^2 a^{-4} -10 z^2+a^3 z-2 z a^{-1} -z a^{-3} +a^4+2 a^2+2
The A2 invariant Data:K11n114/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n114/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a195,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {9_30,}

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{62}{3} -\frac{58}{3} -32 -\frac{112}{3} -\frac{160}{3} 40 -\frac{32}{3} 32 \frac{248}{3} \frac{232}{3} \frac{4529}{30} -\frac{286}{5} \frac{10138}{45} -\frac{1073}{18} \frac{1169}{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=0 is the signature of K11n114. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11         1-1
9        2 2
7       31 -2
5      52  3
3     43   -1
1    55    0
-1   45     1
-3  24      -2
-5 14       3
-7 2        -2
-91         1
Integral Khovanov Homology

(db, data source)

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