From Knot Atlas
Jump to: navigation, search






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

Visit K11n92 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X7,20,8,21 X9,16,10,17 X2,11,3,12 X13,19,14,18 X15,8,16,9 X17,1,18,22 X19,6,20,7 X21,15,22,14
Gauss code 1, -6, 2, -1, 3, 10, -4, 8, -5, -2, 6, -3, -7, 11, -8, 5, -9, 7, -10, 4, -11, 9
Dowker-Thistlethwaite code 4 10 12 -20 -16 2 -18 -8 -22 -6 -14
A Braid Representative
A Morse Link Presentation K11n92 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:K11n92/ThurstonBennequinNumber
Hyperbolic Volume 8.77077
A-Polynomial See Data:K11n92/A-polynomial

[edit Notes for K11n92'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 K11n92's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-3 t^2+3 t-1+3 t^{-1} -3 t^{-2} + t^{-3}
Conway polynomial z^6+3 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 15, -2 }
Jones polynomial -q^4+2 q^3-2 q^2+3 q-2+2 q^{-1} -2 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6-a^2 z^4-z^4 a^{-2} +5 z^4-4 a^2 z^2-3 z^2 a^{-2} +7 z^2+a^4-3 a^2- a^{-2} +4
Kauffman polynomial (db, data sources) a z^9+z^9 a^{-1} +a^2 z^8+2 z^8 a^{-2} +3 z^8-5 a z^7-4 z^7 a^{-1} +z^7 a^{-3} -6 a^2 z^6-11 z^6 a^{-2} -17 z^6+6 a z^5+z^5 a^{-1} -5 z^5 a^{-3} +11 a^2 z^4+17 z^4 a^{-2} +28 z^4-a^3 z^3-3 a z^3+4 z^3 a^{-1} +6 z^3 a^{-3} -a^4 z^2-10 a^2 z^2-9 z^2 a^{-2} -18 z^2+a^3 z+a z-z a^{-1} -z a^{-3} +a^4+3 a^2+ a^{-2} +4
The A2 invariant q^{14}-q^8-q^6+2+ q^{-2} + q^{-4} + q^{-6} - q^{-12}
The G2 invariant Data:K11n92/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {10_136,}

Vassiliev invariants

V2 and V3: (0, 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
0 8 0 -32 -24 0 \frac{80}{3} -\frac{64}{3} 40 0 32 0 0 32 \frac{40}{3} 40 -48 0

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=-2 is the signature of K11n92. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
9       1-1
7      1 1
5     11 0
3    21  1
1  111   1
-1  22    0
-3 11     0
-5 1      -1
-71       1
Integral Khovanov Homology

(db, data source)

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

Back to the top.