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

Visit K11n32 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8493 X5,12,6,13 X2837 X16,9,17,10 X11,6,12,7 X20,14,21,13 X10,15,11,16 X22,17,1,18 X14,20,15,19 X18,21,19,22
Gauss code 1, -4, 2, -1, -3, 6, 4, -2, 5, -8, -6, 3, 7, -10, 8, -5, 9, -11, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 8 -12 2 16 -6 20 10 22 14 18
A Braid Representative
A Morse Link Presentation K11n32 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n32/ThurstonBennequinNumber
Hyperbolic Volume 13.939
A-Polynomial See Data:K11n32/A-polynomial

[edit Notes for K11n32'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 K11n32's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+6 t^2-16 t+23-16 t^{-1} +6 t^{-2} - t^{-3}
Conway polynomial -z^6-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 69, 0 }
Jones polynomial -q^5+3 q^4-6 q^3+10 q^2-11 q+12-11 q^{-1} +8 q^{-2} -5 q^{-3} +2 q^{-4}
HOMFLY-PT polynomial (db, data sources) -z^6+a^2 z^4+2 z^4 a^{-2} -3 z^4+4 z^2 a^{-2} -z^2 a^{-4} -4 z^2+a^4-a^2+3 a^{-2} - a^{-4} -1
Kauffman polynomial (db, data sources) a z^9+z^9 a^{-1} +2 a^2 z^8+3 z^8 a^{-2} +5 z^8+a^3 z^7+3 a z^7+6 z^7 a^{-1} +4 z^7 a^{-3} -a^2 z^6+z^6 a^{-2} +3 z^6 a^{-4} -3 z^6+3 a^3 z^5-9 z^5 a^{-1} -5 z^5 a^{-3} +z^5 a^{-5} +3 a^4 z^4+4 a^2 z^4-10 z^4 a^{-2} -6 z^4 a^{-4} -3 z^4-6 a^3 z^3-8 a z^3-z^3 a^{-1} -z^3 a^{-3} -2 z^3 a^{-5} -4 a^4 z^2-6 a^2 z^2+8 z^2 a^{-2} +4 z^2 a^{-4} +2 z^2+3 a^3 z+5 a z+3 z a^{-1} +2 z a^{-3} +z a^{-5} +a^4+a^2-3 a^{-2} - a^{-4} -1
The A2 invariant Data:K11n32/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n32/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_34, K11n119,}

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

Vassiliev invariants

V2 and V3: (-1, 2)
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 16 8 \frac{34}{3} \frac{14}{3} -64 -\frac{320}{3} -\frac{128}{3} -16 -\frac{32}{3} 128 -\frac{136}{3} -\frac{56}{3} \frac{5489}{30} \frac{862}{15} -\frac{302}{45} \frac{367}{18} -\frac{751}{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 K11n32. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11         1-1
9        2 2
7       41 -3
5      62  4
3     54   -1
1    76    1
-1   56     1
-3  36      -3
-5 25       3
-7 3        -3
-92         2
Integral Khovanov Homology

(db, data source)

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