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

Visit K11n115 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X14,6,15,5 X7,19,8,18 X9,17,10,16 X2,11,3,12 X20,13,21,14 X22,16,1,15 X17,9,18,8 X12,19,13,20 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 K11n115 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n115/ThurstonBennequinNumber
Hyperbolic Volume 14.7634
A-Polynomial See Data:K11n115/A-polynomial

[edit Notes for K11n115'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 K11n115's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+6 t^2-18 t+27-18 t^{-1} +6 t^{-2} - t^{-3}
Conway polynomial -z^6-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 77, 0 }
Jones polynomial 2 q^4-5 q^3+9 q^2-12 q+13-13 q^{-1} +11 q^{-2} -7 q^{-3} +4 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) -z^6+2 a^2 z^4+z^4 a^{-2} -3 z^4-a^4 z^2+3 a^2 z^2-5 z^2+2 a^2+ a^{-4} -2
Kauffman polynomial (db, data sources) a z^9+z^9 a^{-1} +4 a^2 z^8+2 z^8 a^{-2} +6 z^8+6 a^3 z^7+11 a z^7+6 z^7 a^{-1} +z^7 a^{-3} +4 a^4 z^6+2 a^2 z^6+z^6 a^{-2} -z^6+a^5 z^5-10 a^3 z^5-24 a z^5-10 z^5 a^{-1} +3 z^5 a^{-3} -7 a^4 z^4-17 a^2 z^4-z^4 a^{-2} +3 z^4 a^{-4} -14 z^4-a^5 z^3+4 a^3 z^3+15 a z^3+6 z^3 a^{-1} -4 z^3 a^{-3} +3 a^4 z^2+12 a^2 z^2-z^2 a^{-2} -4 z^2 a^{-4} +12 z^2-a^3 z-4 a z-2 z a^{-1} +z a^{-3} -2 a^2+ a^{-4} -2
The A2 invariant Data:K11n115/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n115/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (-3, -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
-12 -8 72 98 38 96 \frac{496}{3} \frac{160}{3} 24 -288 32 -1176 -456 -\frac{10031}{10} \frac{3026}{15} -\frac{13862}{15} \frac{751}{6} -\frac{2191}{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 K11n115. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
9         22
7        3 -3
5       62 4
3      63  -3
1     76   1
-1    77    0
-3   46     -2
-5  37      4
-7 14       -3
-9 3        3
-111         -1
Integral Khovanov Homology

(db, data source)

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

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See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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