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

Visit K11n156 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X3,11,4,10 X12,6,13,5 X14,7,15,8 X16,10,17,9 X11,19,12,18 X22,13,1,14 X20,16,21,15 X17,4,18,5 X2,19,3,20 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 K11n156 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:K11n156/ThurstonBennequinNumber
Hyperbolic Volume 15.3972
A-Polynomial See Data:K11n156/A-polynomial

[edit Notes for K11n156'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 K11n156's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+7 t^2-18 t+25-18 t^{-1} +7 t^{-2} - t^{-3}
Conway polynomial -z^6+z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 77, 0 }
Jones polynomial q^6-4 q^5+7 q^4-10 q^3+13 q^2-13 q+12-9 q^{-1} +6 q^{-2} -2 q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -3 z^4 a^{-2} +z^4 a^{-4} +3 z^4-2 a^2 z^2-4 z^2 a^{-2} +z^2 a^{-4} +6 z^2-a^2- a^{-2} +3
Kauffman polynomial (db, data sources) 3 z^9 a^{-1} +3 z^9 a^{-3} +12 z^8 a^{-2} +6 z^8 a^{-4} +6 z^8+4 a z^7-z^7 a^{-1} -z^7 a^{-3} +4 z^7 a^{-5} +a^2 z^6-35 z^6 a^{-2} -18 z^6 a^{-4} +z^6 a^{-6} -15 z^6-3 a z^5-6 z^5 a^{-1} -14 z^5 a^{-3} -11 z^5 a^{-5} +6 a^2 z^4+33 z^4 a^{-2} +14 z^4 a^{-4} -2 z^4 a^{-6} +23 z^4+3 a^3 z^3+3 a z^3+5 z^3 a^{-1} +11 z^3 a^{-3} +6 z^3 a^{-5} -6 a^2 z^2-12 z^2 a^{-2} -3 z^2 a^{-4} -15 z^2-a^3 z-2 a z-2 z a^{-1} -z a^{-3} +a^2+ a^{-2} +3
The A2 invariant -2 q^{10}+q^8+2 q^6-2 q^4+3 q^2-1+ q^{-2} +2 q^{-4} - q^{-6} +3 q^{-8} -3 q^{-10} + q^{-14} -2 q^{-16} + q^{-18}
The G2 invariant Data:K11n156/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_71, K11n179,}

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

Vassiliev invariants

V2 and V3: (1, 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
4 0 8 \frac{14}{3} -\frac{14}{3} 0 0 32 -32 \frac{32}{3} 0 \frac{56}{3} -\frac{56}{3} \frac{511}{30} \frac{898}{15} -\frac{4738}{45} \frac{641}{18} -\frac{449}{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 K11n156. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
13         11
11        3 -3
9       41 3
7      63  -3
5     74   3
3    66    0
1   67     -1
-1  47      3
-3 25       -3
-5 4        4
-72         -2
Integral Khovanov Homology

(db, data source)

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