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

Visit K11n119 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,5,15,6 X7,20,8,21 X2,10,3,9 X16,11,17,12 X18,14,19,13 X8,15,9,16 X22,17,1,18 X19,6,20,7 X12,22,13,21
Gauss code 1, -5, 2, -1, 3, 10, -4, -8, 5, -2, 6, -11, 7, -3, 8, -6, 9, -7, -10, 4, 11, -9
Dowker-Thistlethwaite code 4 10 14 -20 2 16 18 8 22 -6 12
A Braid Representative
A Morse Link Presentation K11n119 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:K11n119/ThurstonBennequinNumber
Hyperbolic Volume 14.3682
A-Polynomial See Data:K11n119/A-polynomial

[edit Notes for K11n119'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 K11n119'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 -2 q^3+6 q^2-8 q+11-12 q^{-1} +11 q^{-2} -9 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) -a^2 z^6+a^4 z^4-4 a^2 z^4+3 z^4+2 a^4 z^2-8 a^2 z^2-2 z^2 a^{-2} +7 z^2+2 a^4-5 a^2- a^{-2} +5
Kauffman polynomial (db, data sources) 2 a^3 z^9+2 a z^9+4 a^4 z^8+9 a^2 z^8+5 z^8+3 a^5 z^7+a^3 z^7+2 a z^7+4 z^7 a^{-1} +a^6 z^6-11 a^4 z^6-26 a^2 z^6+z^6 a^{-2} -13 z^6-9 a^5 z^5-14 a^3 z^5-10 a z^5-5 z^5 a^{-1} -3 a^6 z^4+8 a^4 z^4+29 a^2 z^4+6 z^4 a^{-2} +24 z^4+7 a^5 z^3+12 a^3 z^3+8 a z^3+6 z^3 a^{-1} +3 z^3 a^{-3} +2 a^6 z^2-4 a^4 z^2-19 a^2 z^2-7 z^2 a^{-2} -20 z^2-2 a^5 z-3 a^3 z-2 a z-2 z a^{-1} -z a^{-3} +2 a^4+5 a^2+ a^{-2} +5
The A2 invariant q^{18}-q^{16}+2 q^{14}+q^{12}-3 q^{10}+q^8-3 q^6+q^2+4 q^{-2} - q^{-4} +2 q^{-6} + q^{-8} -2 q^{-10}
The G2 invariant Data:K11n119/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_34, K11n32,}

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{14}{3} \frac{14}{3} -64 -\frac{416}{3} -\frac{128}{3} -48 -\frac{32}{3} 128 \frac{56}{3} -\frac{56}{3} \frac{11009}{30} -\frac{418}{15} \frac{7618}{45} \frac{1087}{18} \frac{449}{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 K11n119. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7         2-2
5        4 4
3       42 -2
1      74  3
-1     65   -1
-3    56    -1
-5   46     2
-7  25      -3
-9 14       3
-11 2        -2
-131         1
Integral Khovanov Homology

(db, data source)

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

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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