From Knot Atlas
Jump to: navigation, search






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

Visit K11n99 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X5,14,6,15 X7,12,8,13 X9,19,10,18 X2,11,3,12 X13,6,14,7 X15,20,16,21 X17,22,18,1 X19,9,20,8 X21,16,22,17
Gauss code 1, -6, 2, -1, -3, 7, -4, 10, -5, -2, 6, 4, -7, 3, -8, 11, -9, 5, -10, 8, -11, 9
Dowker-Thistlethwaite code 4 10 -14 -12 -18 2 -6 -20 -22 -8 -16
A Braid Representative
A Morse Link Presentation K11n99 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2,3\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n99/ThurstonBennequinNumber
Hyperbolic Volume 11.6344
A-Polynomial See Data:K11n99/A-polynomial

[edit Notes for K11n99's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant 2

[edit Notes for K11n99's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^2+10 t-13+10 t^{-1} -3 t^{-2}
Conway polynomial -3 z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 39, -2 }
Jones polynomial 3 q^{-1} -4 q^{-2} +6 q^{-3} -7 q^{-4} +6 q^{-5} -6 q^{-6} +4 q^{-7} -2 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8+a^8-z^4 a^6-z^2 a^6-2 z^4 a^4-5 z^2 a^4-4 a^4+3 z^2 a^2+4 a^2
Kauffman polynomial (db, data sources) z^6 a^{10}-4 z^4 a^{10}+4 z^2 a^{10}+2 z^7 a^9-7 z^5 a^9+6 z^3 a^9-z a^9+2 z^8 a^8-6 z^6 a^8+5 z^4 a^8-4 z^2 a^8+a^8+z^9 a^7-2 z^7 a^7+4 z^5 a^7-11 z^3 a^7+5 z a^7+3 z^8 a^6-9 z^6 a^6+12 z^4 a^6-9 z^2 a^6+z^9 a^5-4 z^7 a^5+14 z^5 a^5-21 z^3 a^5+11 z a^5+z^8 a^4-2 z^6 a^4+3 z^4 a^4+4 z^2 a^4-4 a^4+3 z^5 a^3-4 z^3 a^3+5 z a^3+5 z^2 a^2-4 a^2
The A2 invariant Data:K11n99/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n99/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_144,}

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

Vassiliev invariants

V2 and V3: (-2, 6)
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 48 32 -\frac{172}{3} \frac{124}{3} -384 -576 -192 -240 -\frac{256}{3} 1152 \frac{1376}{3} -\frac{992}{3} \frac{53849}{15} \frac{1244}{15} \frac{67316}{45} \frac{2935}{9} \frac{329}{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=-2 is the signature of K11n99. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
-1        33
-3       21-1
-5      42 2
-7     32  -1
-9    34   -1
-11   33    0
-13  13     -2
-15 13      2
-17 1       -1
-191        1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}

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.