From Knot Atlas
Jump to: navigation, search






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

Visit K11n132 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X5,19,6,18 X7,16,8,17 X9,14,10,15 X2,11,3,12 X13,20,14,21 X15,8,16,9 X17,1,18,22 X19,12,20,13 X21,7,22,6
Gauss code 1, -6, 2, -1, -3, 11, -4, 8, -5, -2, 6, 10, -7, 5, -8, 4, -9, 3, -10, 7, -11, 9
Dowker-Thistlethwaite code 4 10 -18 -16 -14 2 -20 -8 -22 -12 -6
A Braid Representative
A Morse Link Presentation K11n132 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n132/ThurstonBennequinNumber
Hyperbolic Volume 10.2668
A-Polynomial See Data:K11n132/A-polynomial

[edit Notes for K11n132's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 0
Rasmussen s-Invariant 0

[edit Notes for K11n132's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^2-6 t+9-6 t^{-1} +2 t^{-2}
Conway polynomial 2 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 25, 0 }
Jones polynomial -q+3-3 q^{-1} +4 q^{-2} -4 q^{-3} +4 q^{-4} -3 q^{-5} +2 q^{-6} - q^{-7}
HOMFLY-PT polynomial (db, data sources) -z^2 a^6-a^6+z^4 a^4+2 z^2 a^4+a^4+z^4 a^2+2 z^2 a^2+a^2-z^2
Kauffman polynomial (db, data sources) a^5 z^9+a^3 z^9+2 a^6 z^8+3 a^4 z^8+a^2 z^8+a^7 z^7-3 a^5 z^7-4 a^3 z^7-10 a^6 z^6-13 a^4 z^6-3 a^2 z^6-5 a^7 z^5-a^5 z^5+6 a^3 z^5+2 a z^5+14 a^6 z^4+15 a^4 z^4+a^2 z^4+7 a^7 z^3+4 a^5 z^3-8 a^3 z^3-5 a z^3-6 a^6 z^2-7 a^4 z^2+a^2 z^2+2 z^2-2 a^7 z-a^5 z+3 a^3 z+3 a z+z a^{-1} +a^6+a^4-a^2
The A2 invariant Data:K11n132/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n132/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_8, 10_129, K11n39, K11n45, K11n50,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n50, K11n133,}

Vassiliev invariants

V2 and V3: (2, -3)
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
8 -24 32 \frac{220}{3} -\frac{4}{3} -192 -336 0 -88 \frac{256}{3} 288 \frac{1760}{3} -\frac{32}{3} \frac{20431}{15} -\frac{1444}{15} \frac{23524}{45} \frac{689}{9} \frac{511}{15}

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 K11n132. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
3        1-1
1       2 2
-1      22 0
-3     21  1
-5    22   0
-7   22    0
-9  12     1
-11 12      -1
-13 1       1
-151        -1
Integral Khovanov Homology

(db, data source)

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

Back to the top.