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

Visit K11n157 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X3,11,4,10 X5,12,6,13 X14,7,15,8 X9,16,10,17 X11,19,12,18 X22,13,1,14 X20,16,21,15 X17,4,18,5 X19,3,20,2 X8,21,9,22
Gauss code 1, 10, -2, 9, -3, -1, 4, -11, -5, 2, -6, 3, 7, -4, 8, 5, -9, 6, -10, -8, 11, -7
Dowker-Thistlethwaite code 6 -10 -12 14 -16 -18 22 20 -4 -2 8
A Braid Representative
A Morse Link Presentation K11n157 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:K11n157/ThurstonBennequinNumber
Hyperbolic Volume 14.7471
A-Polynomial See Data:K11n157/A-polynomial

[edit Notes for K11n157'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 K11n157's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+6 t^2-15 t+21-15 t^{-1} +6 t^{-2} - t^{-3}
Conway polynomial 1-z^6
2nd Alexander ideal (db, data sources) \left\{3,t^2+1\right\}
Determinant and Signature { 65, 0 }
Jones polynomial -q^3+4 q^2-7 q+10-11 q^{-1} +11 q^{-2} -9 q^{-3} +7 q^{-4} -4 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) -a^2 z^6+a^4 z^4-3 a^2 z^4+2 z^4+a^4 z^2-3 a^2 z^2-z^2 a^{-2} +3 z^2+1
Kauffman polynomial (db, data sources) 2 a^3 z^9+2 a z^9+5 a^4 z^8+8 a^2 z^8+3 z^8+4 a^5 z^7+2 a^3 z^7-a z^7+z^7 a^{-1} +a^6 z^6-13 a^4 z^6-20 a^2 z^6-6 z^6-11 a^5 z^5-16 a^3 z^5-2 a z^5+3 z^5 a^{-1} -2 a^6 z^4+5 a^4 z^4+13 a^2 z^4+4 z^4 a^{-2} +10 z^4+6 a^5 z^3+9 a^3 z^3+a z^3-z^3 a^{-1} +z^3 a^{-3} +a^6 z^2+a^4 z^2-3 a^2 z^2-2 z^2 a^{-2} -5 z^2+1
The A2 invariant q^{18}-2 q^{16}+q^{14}-2 q^{10}+3 q^8-q^6+2 q^4-1+2 q^{-2} -2 q^{-4} +2 q^{-6} + q^{-8} - q^{-10}
The G2 invariant Data:K11n157/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: (0, 0)
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
0 0 0 0 0 0 0 -32 32 0 0 0 0 0 128 -128 0 -32

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 K11n157. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7         1-1
5        3 3
3       41 -3
1      63  3
-1     65   -1
-3    55    0
-5   46     2
-7  35      -2
-9 14       3
-11 3        -3
-131         1
Integral Khovanov Homology

(db, data source)

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

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