From Knot Atlas
Jump to: navigation, search






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

Visit K11n159 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X12,6,13,5 X7,18,8,19 X9,20,10,21 X16,11,17,12 X22,13,1,14 X4,16,5,15 X2,17,3,18 X19,8,20,9 X14,21,15,22
Gauss code 1, -9, 2, -8, 3, -1, -4, 10, -5, -2, 6, -3, 7, -11, 8, -6, 9, 4, -10, 5, 11, -7
Dowker-Thistlethwaite code 6 10 12 -18 -20 16 22 4 2 -8 14
A Braid Representative
A Morse Link Presentation K11n159 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n159/ThurstonBennequinNumber
Hyperbolic Volume 15.181
A-Polynomial See Data:K11n159/A-polynomial

[edit Notes for K11n159's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 2

[edit Notes for K11n159's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-6 t^2+17 t-23+17 t^{-1} -6 t^{-2} + t^{-3}
Conway polynomial z^6+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 71, -2 }
Jones polynomial -1+5 q^{-1} -8 q^{-2} +11 q^{-3} -12 q^{-4} +12 q^{-5} -10 q^{-6} +7 q^{-7} -4 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8-2 z^4 a^6-3 z^2 a^6-a^6+z^6 a^4+3 z^4 a^4+4 z^2 a^4+a^4-z^4 a^2+a^2
Kauffman polynomial (db, data sources) z^6 a^{10}-2 z^4 a^{10}+z^2 a^{10}+4 z^7 a^9-11 z^5 a^9+7 z^3 a^9+5 z^8 a^8-12 z^6 a^8+4 z^4 a^8+z^2 a^8+2 z^9 a^7+4 z^7 a^7-21 z^5 a^7+13 z^3 a^7-2 z a^7+9 z^8 a^6-21 z^6 a^6+13 z^4 a^6-5 z^2 a^6+a^6+2 z^9 a^5+2 z^7 a^5-9 z^5 a^5+6 z^3 a^5-2 z a^5+4 z^8 a^4-8 z^6 a^4+12 z^4 a^4-6 z^2 a^4+a^4+2 z^7 a^3+z^5 a^3+z^3 a^3+5 z^4 a^2-z^2 a^2-a^2+z^3 a
The A2 invariant q^{28}-q^{26}-2 q^{24}+2 q^{22}-2 q^{20}+q^{18}+q^{16}-2 q^{14}+2 q^{12}-2 q^{10}+3 q^8+q^6-q^4+3 q^2-1
The G2 invariant Data:K11n159/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

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{268}{3} \frac{44}{3} -192 -368 -96 -24 \frac{256}{3} 288 \frac{2144}{3} \frac{352}{3} \frac{22951}{15} \frac{1252}{5} \frac{21124}{45} -\frac{439}{9} \frac{871}{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 K11n159. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
1         1-1
-1        4 4
-3       52 -3
-5      63  3
-7     65   -1
-9    66    0
-11   46     2
-13  36      -3
-15 14       3
-17 3        -3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
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.